2019 AP Calculus AB FRQ #6

Functions f, g, and h are twice-differentiable functions with g(2) = h(2) = 4. The line y = 4 + 2/3 (x - 2) is tangent to both the graph of g at x = 2 and the graph of h at x = 2.
(a) Find h’(2).
(b) Let a be the function given by a(x) = 3*x^3 h(x). Write an expression for a’(x). Find a’(2).
(c) The function h satisfies h(x) = x^2 - 4 / 1 − (f(x)^3) for x does not equal 2. It is known that lim (x →2) h(x) can be evaluated using L’Hospital’s Rule. Use lim (x →2) h(x) to find f(2) and f’(2). Show the work that leads to your answers.
(d) It is known that g(x) is less than or equal to h(x) for x is between 1 and 3. Let k be a function satisfying g(x) is less than or equal to k(x) is less than or equal to h(x) for x is between 1 and 3. Is k continuous at x = 2 ? Justify your answer.
Timestamps
Intro: 00:00
Part a: 00:22
Part b: 01:39
Part c: 03:12
Part d: 07:08