Question

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = x³ - 4x on the interval [-1, 1]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.

Answer 1

f(x) = x^3 - 4x
f'(x) = 3x^2 - 4

First of all, polynomial functions are all continuous in nature, so the function in continuous in [-1,1] and differentiable in (-1,1)

Now,

f(1) = f(a) = -3, f(-1) = f(b) = 3

Now, there exists one number c such that:

f'(c) = (f(b) - f(a))/(b-a)
=> 3c^2 - 4 = (3 - (-3))/(-1-1)
=> 3c^2 - 4 = -3
=> 3c^2 = 1
=> c = ±√(1/3)

Answer 2

Thank you!

Answer 3

that is not the mean value theorem for INTEGRALS