Question

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = x^3 - 9x on the interval [-1, 1]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.
I think one point is zero.

Answer 1

well, f(x) is continuous, so the MVT applies.

(f(1)-f(-1))/(1-(-1)) = (-8-8)/2 = -8

So, you want c where f'(c) = -8

3c^2-9 = -8
c = ±1/?3

Note that at f'(0) = -9, so it is too steep. You were just guessing, there, right? Just to show that our c values work, note that f(x) is odd, so if one works, the other does as well.

http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E3-9x,+y%3D-8(x-1%2F%E2%88%9A3)-26%2F(3%E2%88%9A3)

Answer 2

Thank you!