Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

Question

F(x) = x^3-4x^2-16x+9, [-4,4]

I got to the point of finding the derivative of the function where it equals: F'(x) = 3x^2-8x-16

When solved, I got the solutions of x=-4/3 and x=4. When I submit these answers they are marked incorrect.

Answer 1

Here is a statement of Rolle's Theorem:

"If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f (a) = f (b), then there exists at least one c in the open interval (a, b) such that f′(c)=0."

1. f is a polynomial so it is continuous everywhere and specifically in the closed interval [-4, 4].
2. f is a polynomial so it is differentiable everywhere and specifically in the open interval (-4, 4).
3. f(4) = f(-4) = -55

So the 3 conditions of the Theorem are satisfied.

Your derivative is correct and so are the solutions that you found to f'(x) = 0.

Here is where the issue happens...

Notice that the Theorem guarantees the existence of a c in the open interval (a, b).

So while -4/3 is a value that satisfies the conclusion of Rolle's Theorem,

the fact that the derivative is equal to 0 at 4 is "coincidental", in the sense that it is not a consequence of this particular Theorem.

Answer 2

https://www.youtube.com/watch?v=vRei5wrYgSw

See the video.