Maximum Product of Two Numbers whose sum is 30

This is a classic quadratic problem.
The two numbers are x and 30-x
Their product is (x)(30-x), which is a parabola.
The MAXIMUM of the parabola is the vertex.
You can find it in many ways, here, I use factoring.

[@jakehobrath7721]

d/dx(30x-x^2)=30-2x, a quadratics have a min/max where dy/dx equals zero so 30-2x=0, x =15

[@lechaiku]

This is just optimization problem that can be solved immediately by using the property of area optimization.
That's the reason why among all rectangles of a given perimeter, the square has the largest area.
It means that x = y
So when we have a such kind of problem (maximum product of two numbers) we need just one step.
If you have a sum (x+y) just divide it by 2.
30: 2 = 15 ------------> x = 15 and y = 15
That's it.
Or
when you don't have a sum but you have a product
A = xy
like in
P = x (30 -x)
what you need is only to make a comparison of one factor with other factor
x = 30 -x
2x = 30
x = 15
Also fast and easy.

[@mroldridge]

Yes, you are correct. My experience as a teacher shows that other teachers give questions like these and they want you to solve it as a quadratic equation
But truthfully I use your method in the real world when this is ncessary

[@david7996]

Seems to me like you could also solve this with lagrange?

[@mroldridge]

Probably. Been a long time since I've seen an L

[@BeaDSM]

But where's the proof? Also think it looks nicer to write it as P = (15 + x)(15 - x) as if you don't need to prove it is obvious then by inspection what value of x gives the maximum product (x = 0, so P(max) = 15 × 15 = 225).

[@mroldridge]

It DOES look nicer to write it that way. I have aimed this video at Grade 11 math students that I teach, where they are *just* beginning to look at and understand how to solve quadratic equations. "x" and "30-x" I find are more intuitive for that level. My apologies if you found yourself here from a higher math level.

[@BeaDSM]

@@mroldridge fair enough!

[@howtotutorials8364]

CRAZY!!!!!

[@RameshKR-y6y]

AM-GM inequality