Maximizing 3x+4y Given That x^2+y^2=4

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  • Опубликовано: 24 дек 2024

Комментарии • 304

  • @SyberMath
    @SyberMath  3 года назад +10

    Here is Riley's video on Lambert-W function! Check it out:
    ruclips.net/video/Go5W4Nacn3Q/видео.html

    • @maxt6452
      @maxt6452 3 года назад +1

      I can solve this problem when I put x=2sina, y=2cosa then equation is satisfied. And I put value of x and y in last equation like 6sina+8cosa then Max value is √6^2+8^2=10 . I wait for you your reply. I am a jee student from India🇮🇳

    • @bitsavas
      @bitsavas 3 года назад

      you are teaching maths? in scholl mr sybermath?

    • @cicik57
      @cicik57 2 года назад

      so you take maximum 3x+4y = (3² + 4²) (x²+y²) = 100; then you have system of 2 equasions and can get x and y; for example x = 6/5

  • @BCS-IshtiyakAhmadKhan
    @BCS-IshtiyakAhmadKhan 3 года назад +43

    Since x, y lies on the circle with radius 2 we can write x=2cost,y=2sint now 3x +4y becomes 6cost+8sint and we know that max. value of acosx+bsinx is √(a^2+b^2) so the max. value of our expression will be ✓(6^2+8^2)=10

    • @별의별-h9b
      @별의별-h9b 3 года назад

      I thought so too.
      As soon as I saw the equation, I thought that the traces of x,y were circles.

    • @JohnRandomness105
      @JohnRandomness105 3 года назад

      Definitely more refined than either of my methods.

    • @italixgaming915
      @italixgaming915 3 года назад +2

      I used a similar method but I didn't use trigonometry, just equations of lines:
      x²+y²=4 is the equation of the circle centered on O of coordinates (0,0) and of radius 2.
      3x+4y=K, where K is a real number, is the equation of a line. We're looking at the maximum value of K where the line intersects the circle.
      We can see that when K rises, our lines moves upwards. If K is too low or too high, there are no points of the line that are also on the circle and when K rises inside the range where the line and the circle meet each other, the common points move to the north-east.
      So the value of K that we're looking for is where the line is tangent to the circle in the north-east. But since the equation of the line is 3x+4y=K and since the radius is coming from O, we know that the radius belongs to the line of equation: y=4/3.x.
      We report this into the equation of the circle: 25/9.x²=4 => x²=36/25 => x=6/5 (x is positive).
      y=4/3.(6/5)=8/5.
      Then 3x+4y=18/5+32/5=10.

    • @clayton97330
      @clayton97330 2 года назад

      I made the mistake of brute force. In the circle equation solve for y, substitute into linear equation, set derivative equal to zero, solve for x. Find y, substitute both values into linear equation. Result = 10.

  • @chucktang9113
    @chucktang9113 3 года назад +23

    Using Cauchy inequality:(x^2+y^2)(9+16)≥(3x+4y)^2 , (3x+4y)^2≤100 ,3x+4y≤10 ,so that the maximum is 10

  • @kartikeyagarwal5156
    @kartikeyagarwal5156 3 года назад +60

    Using trigonometric substitution, we can directly see the answer to be 10.

    • @SyberMath
      @SyberMath  3 года назад +5

      How?

    • @MathElite
      @MathElite 3 года назад +24

      @@SyberMath substituting x=2costheta and y=2sintheta leaving us with max(6costheta + 8sintheta) and the sqrt(6^2+8^2) = 10

    • @crazy4hitman755
      @crazy4hitman755 3 года назад +1

      Same

    • @kartikeyagarwal5156
      @kartikeyagarwal5156 3 года назад +5

      @@SyberMath After the substitution, we will get something like acos(A)+bsin(A), whose range is [-sqrt(a²+b²), sqrt(a²+b²)]
      So maximum value is right extremum...

    • @PS-mh8ts
      @PS-mh8ts 3 года назад +13

      @@SyberMath Perhaps something along the following lines:
      Given x²+y²=4
      In polar coordinates,
      x=2cosθ
      y=2sinθ
      Hence 3x+4y=3(2cosθ)+4(2sinθ)=6cosθ+8sinθ -- (i)
      Consider a right-triangle with shorter edges 6 and 8 (the coefficients of cosθ and sinθ in (i)). The length of the hypotenuse = √((6*6)+(8*8)) = 10
      Let's write 6cosθ+8sinθ = 10[(6/10)cosθ+(8/10)sinθ] -- (ii)
      Suppose sinβ=6/10; then cosβ=8/10
      Substituting these in (ii), we get:
      3x+4y=10[sinβcosθ+cosβsinθ]=10sin(β+θ) -- (iii)
      Thus, the maximum of 3x+4y=maximum of 10sin(β+θ) = 10 (because the maximum value of sin(β+θ) = 1)

  • @fantasysco
    @fantasysco 3 года назад +5

    You can also actually draw a circle with the centre being the origin and the radius being 2. Then shift a line whose slope is -3/4 to where the line is tangential to the circle. There should be 2 tangential point, the position of the upper one is the x and y of the answer, substituting them yields the maximum.

  • @InnocentNeuron
    @InnocentNeuron 3 года назад +1

    There is a simpler way of doing this by setting up a Lagrange multiplier, such that L = 3x + 4y - m(4 - x^2 - y^2). Now you partially differentiate L wrt x, y, and m respectively and eliminate m to find the solution.

  • @davidseed2939
    @davidseed2939 3 года назад +2

    Note: x^2+y^2=r^2 is the equation of a circle. It is also the pythagorean theorem. And with 3 and 4 we can see a 3,4,5 triangle coming up.
    3x+4y=c=10 is the equation of the tangent(slope=-3/4). And 4x-3y=0 is the radius that meets the tangent at right angles.. slope=4/3.
    So you could solve it using y=4x/3 substituted into the equation for the circle (25/9)x^2 =2^2 => x=6/5 so y=8/5, 3x+4y=50/5=10

  • @ShefsofProblemSolving
    @ShefsofProblemSolving 3 года назад +2

    A way I tried to visualize this is the following:
    x^2 + y^2 = 4 is a circle with center at (0, 0) and radius 2. Now 4y + 3x = c needs the highest c such that this line will intersect the circle. Now you know this line has to be tangent to the circle and you have two such points which you can test out yourself (as c increases the lines that are formed are parallel to one another if I'm not mistaken)

    • @abustefano8225
      @abustefano8225 3 года назад

      Yeah.
      The geometric way is the quickest, indeed.
      It takes one second to come to the solution: the function f(x, y) :3x+4y =a has the max if a =10 ( tangent the circle)

  • @jayatemihir5390
    @jayatemihir5390 3 года назад +5

    We can also solve by geometry by considering x^2+y^2=4 as a circle in coordinate plane with origin as the centre and as we know that we have to find max(3x+4y)=k, so let's consider a straight line 3x+4y=k and from graph we know as a,b>0, where a,b are coefficients of x and y so we find that k is maximum when the line is at the maximum distance from origin and the slope is negative, so to have maximum value of k for a point on the circle, line must much farther that it must meet the circle at one point which means that k must be that the line touches the circle and the we can use distance of line from origin=radius of circle as origin is the centre of the circle and we can get the max value of k

  • @bjoernschermbach3957
    @bjoernschermbach3957 3 года назад +5

    I argued that the dot product of (3,4) and (x,y) reaches its maximum value when (x,y) is a positive multiple of (3,4). Because of the condition, the length must be 2, so I took (2/5)×(3,4) and put the resulting values into the expression 3x+4y

  • @christopherrice4360
    @christopherrice4360 3 года назад +7

    I've never seen something like this before. It's right up my alley for what i am looking for. The whole concept of elementary algebra concepts "growing up" and transforming into higher Mathematics. I'm excited to watch this video!!

    • @SyberMath
      @SyberMath  3 года назад +2

      Very glad to hear that! ☺️

  • @GreenMeansGOF
    @GreenMeansGOF 3 года назад +14

    For proof that 10 is attainable, use x=6/5 and y=8/5.

  • @leecherlarry
    @leecherlarry 3 года назад +17

    optimization problem, nice:
    *Maximize[{3 x + 4 y, x^2 + y^2 == 4}, {x, y}]*

    • @SyberMath
      @SyberMath  3 года назад +5

      Hello computer! 🤗😂

  • @simplyamazing4219
    @simplyamazing4219 3 года назад +1

    One more approach would be to assume a vector A=3i+4j and B=xi+yj
    A.B=|A| × |B| ×cos∅, where ∅ is the angle between A and B.
    So
    (3i+4y)(xi+yj)=√3²+4² × √x²+y² ×cos∅
    3x+4y=5√4cos∅
    3x+4y=10cos∅. For 3x+4y to be maximum, cos ∅ should be maximum and range of cos ∅ is -1≤cos∅≤1. So (cos ∅)max=1
    So
    (3x+4y)max=10

  • @stingl8822
    @stingl8822 3 года назад +13

    We can also use lagrangian multipliers, right? Or am I wrong?

    • @billprovince8759
      @billprovince8759 3 года назад +3

      Indeed, and you in fact, this was my first thought as well. Having said that, it's interesting to see a different approach.

    • @SyberMath
      @SyberMath  3 года назад +3

      You can!

    • @get2113
      @get2113 3 года назад +1

      Right

    • @panos1435
      @panos1435 3 года назад +1

      Yes, of course! It's easy to find that the function f(x,y)=3x+4y has a maximum on the curve g(x,y)=x^2+y^2-4=0 at the point (6/5,8/5) the value 10. And by symmetry it has a minimum if you reflect the point on the origin. By the way this channel is awesome! It's fun to see different methods!

  • @tgx3529
    @tgx3529 3 года назад +1

    x=r*cos alfa,y=r*sin alfa;there is maximum for r=2; alfa=arctg(4/3)

  • @skylardeslypere9909
    @skylardeslypere9909 3 года назад +2

    What I did was solve x²+y²=4 for y (positive square root), plug it into 3x+4y, differentiate wrt x and set equal to 0. Find x, find y, and compute 3x+4y = 10

  • @Blaqjaqshellaq
    @Blaqjaqshellaq Год назад

    I approached it with polar co-ordinates: r^2=4 and we want to maximize f(theta)=3*r*cos(theta) + 4*r*sin(theta). Firstly, r=+(4)^1/2=2.
    Secondly, f'(theta)=4*r*cos(theta) - 3*r*sin(theta), so the maximum for f should be at the point where 4*r*cos(theta) - 3*r*sin(theta)=0, therefore
    8*cos(theta)=6*sin(theta), therefore
    sin(theta)=(8/6)*cos(theta)=(4/3)*cos(theta), therefore
    theta=Arctan(4/3), at which point sin(theta)=4/5 and cos(theta)=3/5, therefore
    y=r*sin(theta)=2*4/5=1.6 and x=r*cos(theta)=2*3/5=1.2 and f(theta)=4*1.6 + 3*1.2=6.4+3.6=10

  • @gauravsaxena6034
    @gauravsaxena6034 3 года назад +4

    Most of your questions are from H.S haul book, I like this book. The questions are very hard in this. Your solutions are awesome.

    • @SyberMath
      @SyberMath  3 года назад

      Which book is that?

    • @gauravsaxena6034
      @gauravsaxena6034 3 года назад

      @@SyberMath Hall and Knight Higher Algebra book, misspelled in my previous comment. I think you have seen this book. You can google it. Well thanks for liking my comment.

    • @engjayah
      @engjayah 3 года назад

      @@SyberMath I think he meant the famous Text Book called School Algebra by H. S Hall and Night published end of 1890s and many reprints till date.

  • @mathenthusiasts444
    @mathenthusiasts444 3 года назад +1

    Let x = 2*sin a. Then y = 2*cos a.
    Let z = 3x + 4y = 5*(3/5*2*sin a + 4/5*cos a) = 10*sin (a + b) where
    cos b = 3/5 & sin b = 4/5. As the max value of sin(a+b) is one, the max value of z = 10*1 = 10.
    Note that the 5 in the expression for z = square root (3^2 + 4^2) = 5

    • @SyberMath
      @SyberMath  3 года назад

      Wow! That's very cool. Focused so much on this identity, I did not even think about this 😁

    • @mathenthusiasts444
      @mathenthusiasts444 3 года назад

      Your solutions are wonderful. I am a big fan of your videos! Keep posting such wonderful problems & solutions. Thanks.

    • @kristoferescalantejuarez9557
      @kristoferescalantejuarez9557 5 месяцев назад

      De todas las respuestas está me encantó

  • @ishakbelahmar9392
    @ishakbelahmar9392 3 года назад

    Hi, alternative method: if x^2+y^2 = 4 means that (x,y)€circle with radius 2.
    We can then maxmimize : 3*2*cos(t) + 4*2*sin(t) [x=2*cos(t); y=2*sin(t)]
    and we derivate, we have then : 4*cos(t) = 3*sin(t)
    By back substitution : 10*cos(t) ; cos(t) our max is 10

  • @adityaekbote8498
    @adityaekbote8498 3 года назад +6

    Glad I found this channel this is so good.

  • @gtweak7
    @gtweak7 3 года назад +1

    I solved it by expressing the 4y term in terms of x, then equating the derivative of the expression to be maximized to 0 and then observing that for x=6/5 the expression indeed reached the maximum of 10.
    The approach that I used requires the knowledge of how to calculate derivatives, yours with the accompanying reasoning is great.

  • @tamarpeer261
    @tamarpeer261 3 года назад

    Using lazy derivretives:
    y^2=4-x^2
    y=sqrt(4-x^2)
    (3x+4sqrt(4-x^2))’=0
    3-8x/2sqrt(4-x^2)=0
    -4x/sqrt(4-x^2)=3
    16x^2/(4-x^2)=9
    t=x^2
    16t/(4-t)=9
    16t=36-9t
    25t=36
    t=36/25
    x=+-6/5, we of course want it to be positive
    y^2=4-36/25=(100-36)/25=64/25
    y=8/5
    3x+4y=18/5+32/5=10

  • @PS-mh8ts
    @PS-mh8ts 3 года назад

    You can also solve it using vector dot-product as follows:
    3x+4y can be written as the dot-product of vectors (3i+4j) and (xi+yj) where i and j are unit vectors
    but dot product of vectors a and b (i.e,, a.b)=|a||b|cosθ where θ is the angle between the vectors
    Thus, for any two vectors a and b, the dot product is maximum when the angle between them = 0 (because cosθ assumes its maximum 1 when θ=0)
    Thus, maximum value of a.b = |a||b|
    Thus, the maximum value of 3x+4y = maximum value of (3i+4j).(xi+yj) = |3i+4j||xi+yj|
    but |3i+4j|=√(3²+4²)=5
    and |xi+yj|=√(x²+y²)=√4=2 (because it's given that x²+y²=4)
    Thus, the maximum value of 3x+4y = 5 times 2 = 10

  • @felipelopes3171
    @felipelopes3171 3 года назад

    Geometry solves it quickly. The locus of the points satisfying x^2+y^2=4 is just a circle of radius 2 centered on the origin. The locus of points where 3x+4y is a constant are lines with slope -3/4. Maximizing this is to simply pick a line that's furthest to the right.
    Therefore, you are looking for the point in the first quadrant of the circle whose tangent has a slope of -3/4. With geometry you can find that the ray from the origin to this point has tangent 4/3, so it must be (6/5,8/5), and the maximum is 10.

  • @ilhantezel6125
    @ilhantezel6125 3 года назад

    Maximim increase is gradient direction that is 3ux+4uy vector, this direction intersect circle at x=1.2 and y=1.6, substitute these on 3x+4y, you get 10

  • @geosalatast5715
    @geosalatast5715 3 года назад

    Since we know that x^2+y^2=4 (1) is a circle with the origin being the center and radius=2, then the sum 3x+4y (2) can maximize only for values of x and y of the first quadrant. So x,y>0. After that we substitute the value of y to (2) and it becomes 3x+4*sqrt(4-x^2) and using calculus we let f(x)=3x+4*sqrt(4-x^2), Df=[0,2]. We find the first derivative to point out possible maxima where the first derivative becomes zero. This happens at x=3/5. We can check if this is a maximum by finding the sign of the first derivative or by finding the second derivative which is negative so f is concave down. We find the maximum straight from calculating the value of f at x=3/5 or we find the value of y from (1) which is y=8/5 and we substitute the two values in the (2). Both of the solutions give us 10.

  • @laokratis55
    @laokratis55 3 года назад

    Besides the algebraic and the trigonometric there is one more visualization of this problem. If the maximum is m, 3x+4y=m then this is obtained when the line y=-3x/4+m/4 with slope -3/4 is tangent of the circle. Differentiating x^2+y^2=R^2 we get xdx+ydy=0, the slope of the tangent to the circle at any point is dy/dx=-x/y=-3/4 and that is it.

  • @hitesh9997638184
    @hitesh9997638184 3 года назад +2

    There's one more way to solve it, that's using the parametric equation of circle Which leads to an expression of 6cos(a)+8sin(a) which can have a maxima of (6^2+8^2)^0.5

  • @agentbinodbollywoodwale6656
    @agentbinodbollywoodwale6656 3 месяца назад

    This problem can also be solved using quadratic equations. So let's assume 3x + 4y = k(some constant real number), where x^2 + y^2 = 4, so we can write y = (3 / 4)x - (k / 4). Substituting this in the equation x^2 + y^2 = 4, we finally get 25(x^2) - 6kx + (k^2 - 64) = 0. Since, x is the unknown real number and k is the constant, solution of the equation being real or complex depends upon the discriminant D >= 0, so we have a = 25, b = -6k, and c = k^2 - 64. Therefore, b^2 - 4ac >= 0 means (-6k)^2 - 4 * 25 * (k^2 - 64) >= 0 or -64(k^2) + 6400 >= 0 or k^2

  • @vishalmishra3046
    @vishalmishra3046 3 года назад

    x^2 + y^2 = 4 is a 2-unit circle, so x = 2cosT and y = 2sinT such that x^2+y^2 = 4(cos^2 T + sin^2 T) = 4. Now 3x + 4y = 6cosT + 8sinT = 10 sin(T + arcTan(6/8)). Max of sin or cos = 1. So, Max (3x+4y) = 10 x (max of sin or cos) = 10 x 1 = 10. Hence, max (3x+4y) = *10* . Simple, right ?

  • @kaslircribs5804
    @kaslircribs5804 3 года назад

    Your method of solving a problem is amazing. It requires a lot of knowledge and reasoning. Thank you.

  • @orangeguy5463
    @orangeguy5463 3 года назад

    A lot of people are talking about lagrange multipliers. Here's another approach that only uses calc 1: turn it into a single variable optimization with trigonometry by parameterizing x=4cos, y=4sin, and then take the derivative of 12cost + 16sint, look for zeros of 16cost-12sint, so tant=4/3, and substitute back in to x=4cost y=4sint using trigonometric identities to get 2 solutions for max and min. Second derivative test then proves which one is max

  • @miro.s
    @miro.s 3 года назад

    This is also part of linear algebra or quadratic forms and linear transformation. You are linearly transforming circle into ellipse (both eigenvalues are positive), then the square of maximum has meaning as the sum of squares of eigenvalues or also some of squares of all entries in transformation matrix. That is square of radius of Director circle of the ellipse.

  • @MrLeith1975
    @MrLeith1975 3 года назад +1

    Hi there ! Thanks for your channel. It keeps me in touch with mathematics even though i am far away from it ! We can alse use Lagrangien optimisation. we ´ll then find that maximum are reached for x=6/5 and y=8/5 then 3x+4y will give us 10

  • @arundhatimukherjee
    @arundhatimukherjee 3 года назад +1

    Consider the family of parallel straight lines 3x+4y+c=0. x and y lie simultaneously on the line and the circle x²+y²=4, centered at origin. The perpendicular distance of the line from the origin must be ≤ 2 units for the line to intersect, or at least, 'touch' the circle. Mathematically, |c|/5 ≤ 2 or |3x+4y|≤10 or -10≤3x+4y≤10.

    • @kartikeyanand9617
      @kartikeyanand9617 3 года назад +1

      why did you considered family of straight lines

    • @yashjaiswal7438
      @yashjaiswal7438 3 года назад +1

      @@kartikeyanand9617 to make question more easy.. you just have to imagine how line will move on axis... You will find at point of tangency only max value can be found..

    • @SyberMath
      @SyberMath  3 года назад +1

      Good thinking! 😍

    • @arundhatimukherjee
      @arundhatimukherjee 3 года назад

      @@SyberMath Thanks!!!!!

    • @arundhatimukherjee
      @arundhatimukherjee 3 года назад

      @@kartikeyanand9617 Actually when I saw 3x+4y, which is a linear expression in x and y, I thought of straight lines. Family of straight lines makes things easy. 🙂

  • @ahmadkalaoun3473
    @ahmadkalaoun3473 3 года назад +4

    A quick application of the caughy Schwartz inequality would have done it but your way is more elegant :)

    • @falknfurter
      @falknfurter 3 года назад +1

      Well seen! Cauchy-Schwarz inequality is certainly a tool much more powerful than what is required here. But it makes the whole affair basically a one liner ...

    • @ryderpham5464
      @ryderpham5464 3 года назад +1

      If anyone needs a visual of this:
      By Cauchy-Schwarz, we have
      (3^2+4^2)(x^2+y^2) = 25*4 ≥ (3x+4y)^2
      → 10 ≥ 3x+4y, and equality is achieved when x/3=y/4, hence the answer is 10.

  • @richardryan5826
    @richardryan5826 3 года назад +6

    You present an interesting solution. I used the Lagrange multiplier method that's introduced in a third semester Calculus course.

    • @juliocjacobo
      @juliocjacobo 2 года назад +1

      I used Lagrange too. Obtained the same result.

  • @Vladimir_Pavlov
    @Vladimir_Pavlov 3 года назад

    1. You can express y in terms of x: y= ±sqrt (4-x^2) and investigate the function of one variable f(x)=3*x± 4* sqrt (4 - x^2) for extremes, provided -2≤x≤2.
    2. You can use the method of indeterminate Lagrange multipliers - this is the main method for solving such problems. Without any trick

  • @JohnRandomness105
    @JohnRandomness105 3 года назад

    I did the problem using Lagrange multipliers, although solving for y in terms of x, then maximizing by setting the derivative equal to zero would have been just as simple. The maximum value is 10.
    Actually, try it the second way. y = sqrt(4 - x²) ==> dy/dx = -x/y. 0 = 3 + 4 dy/dx = 3 - 4x/y ==> 3y = 4x.
    4 = x² + y² = x² + (4x/3)² = (9x² + 16x²)/9 = 25x²/9. x² = 36/25, x = 6/5. y = 4x/3 = 8/5.
    3x + 4y = (18 + 32)/5 = 10

  • @pedrojose392
    @pedrojose392 3 года назад

    Good evening! I am from Brazil. I notíced tath the purpose of solve this problem is not to use calculus. So I get 3x+4y - C=0 and we want to maximise C with the restriction x^2+y^2=4. So we want the Max C such the line 3x + 4y -C =0 intercepts the circunference x^2+y2=4. This occurs when the line tangents the circunference. So the distance from (0,0) to the line is equal to 2. Then |C|/5=2 and C= + - 10. So C=10 is the maximum.

    • @SyberMath
      @SyberMath  3 года назад

      Good to see you! Greetings from the United States 💖

    • @pedrojose392
      @pedrojose392 3 года назад

      @@SyberMath, I liked the way you solve.

  • @Macialao
    @Macialao 3 года назад

    Much niecr solution can be obtained graphically, if we get 3x+4y = z we can get straight line equation y = -3/4x + z/4 and we know we want to maximize z. So the question is what is the maximum value of z that the line still has intersection with a circle. Or maybe what are extreme cases, when the line has only one intersection point with a circle (at z=-10 or z= 10)

  • @benjaminvatovez8823
    @benjaminvatovez8823 3 года назад

    Thanks for the solution. I did this way: we want to find out m such that the circle x²+y²=4 and the line y=(-3x+m)/4 intersect. Consequently, m,x,y>0 and the line is tangent to the circle and intersects it in only one point, thus the discriminant of x^2+((-3x+m)²/16)=4 equals zero. The rest is calculation.

  • @txikitofandango
    @txikitofandango 3 года назад

    @SyberMath I appreciate that you did it without trig or irrational numbers. Here's how I did it:
    Parameterize the circle as (x,y) = 2/(1+t^2) * (1 - t^2, 2t).
    Maximize P = 3x + 4y which equals (3(1-t^2) + 4(2t))(2/(1+t^2)), which simplifies to P(t) = (-6t^2 + 16t + 6)/(1+ t^2).
    Set P' = 0, and simplify down to 2t^2 + 3t - 2 = 0. That factors as (2t - 1)(t + 2) = 0.
    So t = 1/2 or t = -2
    Plug both values into P and see which is bigger.
    P(1/2) = 10; P(-2) = -10. So, the max is 10.

    • @SyberMath
      @SyberMath  3 года назад

      I don't get how you parametrize the circle 🤔

    • @txikitofandango
      @txikitofandango 3 года назад

      It's based on the Pythagorean formula in one variable:
      (1-t^2)^2 + (2t)^2 = (1+t^2)^2
      Divide both sides by the right-hand side.
      Then match it up with x^2 + y^2 = 1. And then scale (x,y) accordingly with the radius.

    • @txikitofandango
      @txikitofandango 3 года назад

      @@SyberMath ruclips.net/video/xp0H3Aw0j6E/видео.html Wildberger explains it very well

    • @WallaceChan1
      @WallaceChan1 3 года назад

      @@SyberMath he is using half angle substitution t=θ/2 and express sinθ, cosθ in terms of t (x,y)= (2cosθ, 2sinθ) in the equation of the circle

  • @sankarsanbanerji3457
    @sankarsanbanerji3457 3 года назад

    Consider two vectors a= 3i+4j and b= xi+yj. Consider dot product and maximize when cost =1

  • @MalavDoshiDGDKT
    @MalavDoshiDGDKT 3 года назад +1

    Let x=2sinA and y=2cosA then eqn would be 6sinA+8cosA so largest is 10 and lowest -10

  • @MathElite
    @MathElite 3 года назад +3

    What an interesting problem! You are a genius at finding great problems
    I used x=2costheta and y=2sintheta to solve this one!!

  • @catname8813
    @catname8813 3 года назад

    Could we do it like this? Let k=3x+4y, solving for y we get y=(k-3x)/4. Substituting for y in the original equation x^2+y^2=4, we get that x^2+(k^2-6kx+9x^2)/16 = 4. Multiplying the whole thing by 16, we get 16x^2+k^2-6kx+9x^2=64 and if we simplify we get 25x^2-6kx+k^2-64=0. Assuming x is real, we need the discriminant to be nonnegative so we have that 36k^2-4(25)(k^2-64)>=0 or -64k^2+6400>=0. We factor to get -64(k^2-100) >=0 and divide by -64 to get k^2-100

  • @elmatadordeangel5004
    @elmatadordeangel5004 3 года назад

    From x^2 + y^2 = 4, first we can get y from it, which is sqrt(x^2 - 4), substitute it into 3x+4y, and get 3x + 4(sqrt(x^2 - 4)), get the derivative of that and get the critical points by setting it to zero.
    And when we're done we get x= 6/5 or x = -6/5
    We do the same for y, aka we get x from the first equation, and then substitute it into the second one, and we get:
    y = 8/5 or y = -8/5
    Now since we're adding and multiplying by positive numbers, the higher X and Y are, the higher the final value is. Therefore the maximum is X = 6/5 and Y = 8/5
    And when we substitute those values into the equation we get 10
    Solved using calculus

  • @wolfram3319
    @wolfram3319 3 года назад

    Alternative quick solution:
    Let u and v be real vectors. Then,
    max(u•v) = |u|*|v|.
    This is simply true from the definition of the dot product: a•b = |a|*|b|*cos(θ). Clearly, for the dot product to be maximized, cos(θ) must be 1 (i.e., the two vectors must be parallel and are facing the same direction).
    That fact can be used for the problem by letting u be a vector that represents a point on a circle (which in this case, has a radius of 2), and let v be the vector [3,4]. So, we see that
    max(u•v) = 2*5 = 10.
    Problem solved!

  • @15121960100
    @15121960100 6 месяцев назад

    the maximum value occurs at 4y=3x in which case x^2 +y^2 =4 need not necessarily hold good.
    Its value can not cross 100 it doesnt mean it can touch 100.

  • @abhiruppaul5601
    @abhiruppaul5601 3 года назад +1

    Using the graphical solutions for this I got that the maximum value of x is 2 and the maximum value of Y is also 2 so I plugged that in the expression
    3 X + 4 Y to get its maximum value and I got 14 so what do you think about that answer.

    • @axelnils
      @axelnils 3 года назад

      X and Y can’t both be 2, remember x^2 + y^2 = 4. 2^2 + 2^2 = 8 =/= 4.

  • @anaksy4real241
    @anaksy4real241 2 года назад

    Simply awesome. Thanks for the fun and simplicity you brought to the steps.

    • @SyberMath
      @SyberMath  2 года назад

      Np. Thank you for the kind words! 💖

  • @sawyerw5715
    @sawyerw5715 3 года назад

    I intuit that multiplying circle coordinates by coefficients will be a maximum when each coordinate result is equivalent. Thus if 3x=4y, it quickly results in the values for x, y (6/5,8/5) and the result is 10. In essence, you proved this fact in the algebraic property of the sum of squares. Found this interesting to think about.

  • @gal-zki
    @gal-zki 3 года назад +2

    Ok, that was very impressive. However there's one thing that I don't get: I know (4x - 3y)² in the end can't be negative, but isn't it necessary to prove its lowest value is, in fact, 0?
    Could someone help me?

    • @MathElite
      @MathElite 3 года назад

      no you don't have to prove that r^2 >= 0, given that r is a real number, it's a fact

    • @falknfurter
      @falknfurter 3 года назад +1

      You are correct. It's not very hard to show: Smallest value of (4x-3y)^2 is 0. From 4x-3y=0 follows y=4x/3. Substitute this into x^2 + y^2 = 4 and you get x=6/5 and also y=8/5. Negative solutions for x, y can be disregared because they do not provide the maximum.

    • @gal-zki
      @gal-zki 3 года назад

      @@MathElite thank you, but I was actually talking about this particular case (4x - 3y)²

    • @gal-zki
      @gal-zki 3 года назад

      @@falknfurter got it. Thank you very much

    • @anshumanagrawal346
      @anshumanagrawal346 3 года назад

      Well, the answer is yes AND no, technically you should prove it. But it's clear that it would be zero when x and y are in the ratio of 3:4, and we can see that there must exist some values which satisfy that AND x^2+y^2=4 as it is essentially just the equation of a circle

  • @HemantPandey123
    @HemantPandey123 3 года назад

    Alternately we might have directly used Cauchy's inequality. abs val (a.b)

  • @vasilerotaru439
    @vasilerotaru439 3 года назад

    I used CBS inequality (ax+by)^2

  • @davidjames1684
    @davidjames1684 3 года назад

    The ratio of x to y for the answer is 3:4 so is that a coincidence or is it related to 3x+4y that we are trying to maximize? If it is related, then we have more information. I wrote a computer program checking all values for x and y starting at 0 and incrementing by 0.01, but if I knew the answer to x and y was in a 3:4 ratio, I could instead make y always equal to 4/3rd that of x.

  • @shoryaprakash4528
    @shoryaprakash4528 3 года назад

    Using vectors
    a(v) =3i +4j, mag(a) =5
    b(v) =xi+yj, mag(b) =√x^2+y^2=2
    a(v) .b(v)

  • @UttamKumar-kt4jp
    @UttamKumar-kt4jp 3 года назад

    Suppose (x,y) be a point on a circle of radius 2 put x=2cos@ and y=2sin@ and that will bring you to the end .

  • @johnnath4137
    @johnnath4137 2 года назад

    Set sinu =3/5, cosu = 4/5, cos v = x/2, sinv = y/2 → max(3x/10 + 4y/10) = max(sinucosv + cosusinv) = max(sin(u + v)) = 1 → max(3x + 4y) = 10 max(sin(u + v)) = 10.

  • @danilonascimentorj
    @danilonascimentorj 3 года назад

    I used the cauchy-schwarz inequality to get (a1b1+a2b2)^2

  • @snejpu2508
    @snejpu2508 3 года назад

    Interesting formula and, as always, elegant solution. I prefered something more obvious: f(x)=3x+4sqrt(4-x^2), f'(x)=3+4*(-2x)/2sqrt(4-x^2)=0, 3-4x/sqrt(4-x^2)=0, 3=4x/sqrt(4-x^2), 4x=3sqrt(4-x^2), 16x^2=36-9x^2, x^2=36/25, x=+/-6/5. y=sqrt(4-36/25)=sqrt(64/25)=+/-8/5. Taking both postive values we are left with 3*6/5+4*8/5=18/5+32/5=50/5=10. Technically, it's a little illegal, because using the square root, I use only a half of the domain for x and y. But in this case, positive values give us a bigger value of 3x+4y, so I got away with it.

    • @SyberMath
      @SyberMath  3 года назад

      Good method! Good thinking! 💖

  • @debsuryade2481
    @debsuryade2481 3 года назад

    It will be maximum when the line will be tangent to the circle. Hence, 10 is the answer obviously.

  • @miro.s
    @miro.s 3 года назад

    Very nice example, it has geometrical meaning of defining radius of the Director circle of ellipse defined by semiaxis a and b and also it represents energy of two dimensional oscillator or coupled oscillators of the same frequency. It is also representing amplitude of orthogonally combined cosine and sine of the same frequency/speed factor in Fourier series.

  • @aashsyed1277
    @aashsyed1277 3 года назад +2

    I SWEAR THIS IS SO UNDERRATED

  • @lukastillman7095
    @lukastillman7095 3 года назад +2

    This was the first problem presented by you on this channel that I was able to solve! I used calculus. Thank you very much for your content!

    • @SyberMath
      @SyberMath  3 года назад

      Glad it helped!

    • @anshumanagrawal346
      @anshumanagrawal346 3 года назад

      I used the fact the given condition is the equation of a circle and used some analytical geometry, but I'm curious how you used calculus to solve this

    • @geosalatast5715
      @geosalatast5715 3 года назад

      @@anshumanagrawal346 Since we know that x^2+y^2=4 (1) is a circle with the origin being the center and radius=2, then the sum 3x+4y (2) can maximize only for values of x and y of the first quadrant. So x,y>0. After that we substitute the value of y to (2) and it becomes 3x+4*sqrt(4-x^2) and using calculus we let f(x)=3x+4*sqrt(4-x^2), Df=[0,2]. We find the first derivative to point out possible maxima where the first derivative becomes zero. This happens at x=3/5. We can check if this is a maximum by finding the sign of the first derivative or by finding the second derivative which is negative so f is concave down. We find the maximum straight from calculating the value of f at x=3/5 or we find the value of y from (1) which is y=8/5 and we substitute the two values in the (2). Both of the solutions give us 10.

    • @anshumanagrawal346
      @anshumanagrawal346 3 года назад

      @@geosalatast5715 Oh, I thought of solving this with maxima/minima as well but thought it was too long. When the original comment said they used calculus I thought it was some sort of some directly find the maximum of 3x+4y using partial derivatives

  • @鈴木悠真-n6z
    @鈴木悠真-n6z 3 года назад +2

    Bravo!
    This skill is just beautiful and fantastic!
    You relight my hope of number theory!!!!!

  • @paultoutounji3582
    @paultoutounji3582 3 года назад

    Please, could you solve this problem using the derivatives ? Thank you.

  • @Raynover
    @Raynover 3 года назад

    A more geometric approach:
    Let's consider a point M(x,y). Obviously, the point M lies on the circle with center at the origin and radius 2. We want to find the maximum value of 3x+4y. Let 3x+4y=a. This equation represents a straight line in the xy plane. The point M lies on the line, too. So we want the circle and line to have common points. That happens if-f the distance of the center from the line is less than or equal to the radius. From the formula of the distance between a point and a line we get |a|≤10. So the maximum value is 10. It is achieved when the line (for a=10) is tangent to the circle, that is at the point (6/5,8/5).

  • @gaussiit4855
    @gaussiit4855 3 года назад +1

    This can be done quickly using graphs or trigonometric parametric substitution

    • @dadada6192
      @dadada6192 3 года назад +1

      You were sent to attack the Graphs,not join them.

    • @gaussiit4855
      @gaussiit4855 3 года назад +2

      @@dadada6192 haha lol😂

    • @dadada6192
      @dadada6192 3 года назад +1

      @@gaussiit4855 I liked ur comment.

  • @techysubham1939
    @techysubham1939 3 года назад +1

    Can you do some permutations problems also? That will be fun

  • @archilarkania7203
    @archilarkania7203 3 года назад

    I solved this with analysis - get x from that circle without +/- because during f'(x) = 0 you will be turning it into square anyway and get the same results. You will get y = +/- 8/5 . I'm trying to find algebraic or geometric solutions, but when i know easier way my brain stops working.

  • @holyshit922
    @holyshit922 3 года назад

    x=2cost, y=2sin(t)
    6cos(t)+8sin(t)=10(0.6cos(t)+0.8sin(t))
    =10cos(t - theta)
    cos(theta) = 0.6
    sin(theta) = 0.8
    max = 10 for angle t=theta+2kπ or t=-theta+2kπ where k in Z

    • @MarcoMate87
      @MarcoMate87 3 года назад

      Why also t=-theta+2kπ ?

    • @holyshit922
      @holyshit922 3 года назад

      @@MarcoMate87 generally cos is even so we have to include both series of solution but here we have
      t-theta = 0 + 2kπ and you are right

  • @TheEricthefruitbat
    @TheEricthefruitbat 3 года назад

    Very elegant. But why run through the specific calculation after you have just proved the general case? Just substitute for a and b in the general formula. Are you trying to make the video longer? ;)

  • @laokratis55
    @laokratis55 3 года назад

    I appreciate this algebric trick but here is a derivation using trigonometric identites. As you mention in another response to a comment it boils down to maximizing (6costheta + 8sintheta), but this also reduces to maximizing cos(theta)+tan(phi)*sin(theta) where phi is the angle with tangent 4/3. Then this can be factored to read max(sin(phi+theta)) which is maximum for phi+theta=pi/2 and this means that tan(theta)=cot(phi)=3/4=x/y as in the condition you obtained algebrically and in some way justifies the inversion of the coefficients.

  • @yoav613
    @yoav613 3 года назад +1

    This problem is not hard to solve by calculus, but that trick is really nice!!

  • @mamurjonuzoqov4587
    @mamurjonuzoqov4587 3 года назад

    If the sample provided for what would have been called the minimum value of the solution. I'm from Uzbekistan,

  • @italixgaming915
    @italixgaming915 3 года назад

    My solution (of course extreeeeeeeeeemely faster):
    Sometimes I use algebra to solve a geometrical problem, this time I do the opposite.
    x²+y²=4 is the equation of the circle centered on O of coordinates (0,0) and of radius 2.
    3x+4y=K, where K is a real number, is the equation of a line. We're looking at the maximum value of K where the line intersects the circle.
    We can see that when K rises, our lines moves upwards. If K is too low or too high, there are no points of the line that are also on the circle and when K rises inside the range where the line and the circle meet each other, the common points move to the north-east.
    So the value of K that we're looking for is where the line is tangent to the circle in the north-east. But since the equation of the line is 3x+4y=K and since the radius is coming from O, we know that the radius belongs to the line of equation: y=4/3.x.
    We report this into the equation of the circle: 25/9.x²=4 => x²=36/25 => x=6/5 (x is positive).
    y=4/3.(6/5)=8/5.
    Then 3x+4y=18/5+32/5=10.

  • @advaykumar9726
    @advaykumar9726 3 года назад +3

    Divide the equation by 4 both sides, then write it as (x/2)^2+(y/2)^2=1 then let x/2=sin theta so y/2=cos theta. Sub x=2sin theta and y=2 cos theta in the 2nd equation we get max(6sin theta +8cos theta). And, max and min value of a sin theta+ b cos theta is ± √(a^2+b^2) So √(6^2+8^2)=√100=10

    • @SyberMath
      @SyberMath  3 года назад

      Pretty interesting!

    • @鈴木悠真-n6z
      @鈴木悠真-n6z 3 года назад

      Why max and min is +or- √[a^2+b^2] ? Can we prove it by subsidiary angle?

    • @advaykumar9726
      @advaykumar9726 3 года назад

      @@鈴木悠真-n6z Yes, you can also do it by taking the derivative and equal it to 0

    • @advaykumar9726
      @advaykumar9726 3 года назад

      And then substitute the value of sin theta and cos theta

    • @鈴木悠真-n6z
      @鈴木悠真-n6z 3 года назад

      @@advaykumar9726 thank you!

  • @kimsanov
    @kimsanov 3 года назад

    How to analytically measure exact value of max?

  • @nawusayipsunam1643
    @nawusayipsunam1643 3 года назад +1

    This is new and nice sol.

  • @manojsurya1005
    @manojsurya1005 3 года назад

    Can anyone explain the derivative approach to this problem ?

  • @ramaprasadghosh717
    @ramaprasadghosh717 3 года назад

    put sin(z) = x/2 and cos(z) = y/2
    then 3 sin(z) + 4 cos(z)
    = 5 sin(z+ arc tan ( 4/3))
    so max ( 3x+4y)
    =2 max ( 10 sin(z+ arc tan ( 4/3))
    = 10

  • @janindukaveesha2467
    @janindukaveesha2467 3 года назад +1

    This also can be solved using differentiation

  • @engjayah
    @engjayah 3 года назад

    Most of the time you are taking a very lengthy approach to solve this kind of easy problems. I think taking an easy approach will make the students interesting.
    Consider the family of parallel lines 3x + 4y = c; the idea is to maximize the value c.
    there will be two parallels in this family tangential to the circle x2 + y2 = 4 and giving maximum value of c which will be 10.

  • @tarunmnair
    @tarunmnair 3 года назад

    Nice, and that is gonna happen at x=1.2 and y=1.6 ... right ?

  • @chennebicken372
    @chennebicken372 3 года назад

    I find your videos great, but it would be very helpful for me, if you tried to explain your plan before giving „random“ identities. I did not really get the sense of the identity before the end.
    Thanks for helping me to get a better intuition on algebraic problem-solving!

  • @anushkrajbordia1873
    @anushkrajbordia1873 3 года назад +1

    Easy trig. substitution or by method of vectors, this q. can be solved. But your approach is formal & leaves no scope for any domain kind of a mistake! Congrats!!!

  • @siddhantsinha9276
    @siddhantsinha9276 3 года назад +1

    I did it orally using cauchy schwarz

  • @呂永志
    @呂永志 3 года назад

    1. Cauchy's inequality.
    2. x= sin t, y= cos t.
    3. Caculus.

  • @satyapalsingh4429
    @satyapalsingh4429 3 года назад

    Marvellous method of solving

  • @davidbrisbane7206
    @davidbrisbane7206 3 года назад

    Let's solve this problem using Lagrange multipliers.
    Let F(x,y) = 3x + 4y, subject to the constraint G(x,y) = x^2 + y^2 - 4 = 0.
    Fx = λGx & Fy = λGy, by theorem finds the extreme points, where λ is real and
    Fx is the partial derivative of F wrt to x,
    Fy is the partial derivative of F wrt to y,
    Gx is the partial derivative of G wrt to x,
    Gy is the partial derivative of G wrt to y.
    Now Fx = 3 & Fy = 4 and Gx = 2x & Gy = 2y.
    So, 3 = λ(2x) & 4 = λ(2y)
    So, 9 = 4(λ^2)(x^2) & 16 = 4(λ^2)(y^2)
    So, 25 = 4(λ^2)[x^2 + y^2]
    So, 25 = 4(λ^2)[4], as x^2 + y^2 = 4
    So, 25 = 16(λ^2)
    So, λ = ±5/4 leading to (x,y) = (±6/5, ±8/5) as the extreme points, using 3 = λ(2x) & 4 = λ(2y), which results in the maximum of 3x + 4y being 10 when (x,y) = (6/5, 8/5), and the minimum of
    3x + 4y being -10 when (x,y) = (-6/5, -8/5)

  • @dadada6192
    @dadada6192 3 года назад +3

    Imma solve this mah self.

    • @SyberMath
      @SyberMath  3 года назад

      Cool!

    • @dadada6192
      @dadada6192 3 года назад

      @@SyberMath And I didn't feel like it.

  • @tmacchant
    @tmacchant 3 года назад

    The problem can be easily solved using the method of Lagrange multiplier.

  • @rex_yourbud
    @rex_yourbud 2 года назад +1

    I love this comment section. Such love for Maths. I just used calculus to solve it. 🙂

    • @SyberMath
      @SyberMath  2 года назад

      Glad to hear that! 🥰

  • @tonyhaddad1394
    @tonyhaddad1394 3 года назад +1

    Hii mst syber !! As always amazing problems , until now i didn t watch your approach , i think my approach is more complicated than your s
    I let y=k.x then let 3x + 4y = t
    Now we must maximize t if you plug the substitution in the equations you get a system ..... finaly you get a function of k t(k)=......
    You take the derivitive and you get one solution which is max
    ......... finaly you get t or (3x+4y) =10

    • @SyberMath
      @SyberMath  3 года назад +1

      Hi Tony! Your method is awesome! 🤩

    • @tonyhaddad1394
      @tonyhaddad1394 3 года назад

      @@SyberMath thank uu !!!!!!

  • @hsjkdsgd
    @hsjkdsgd 3 года назад

    Nice problem and solution. Objective fnc 3x+4y can be represented as A*sin(something). Max value is A which in our case is 10.

  • @roberttelarket4934
    @roberttelarket4934 3 года назад

    Absolutely beautiful and clever!!!
    Give me more!!!