Mean Value Theorem Problem

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.

Answer 1

The Mean Value Theorem states that the function is defined and continuous in closed interval, and differentiable in the open interval.

 

This function is continuous.  Therefore, it is defined at all points in the closed interval.

 

The slope between the closed ended points is the same as the derivative at some number c.    -1 < c < 1

 

 

f(-1) = -1 + 9 = 8

f(1) = 1 - 9 = -8

 

f'(c) = (-8 - 8) / (1 - (-1)) = -16 / 2 = -8

 

f'(c) = -8

 

 

Set the derivative of f(x) equal to 8.

 

3x² - 9 = -8

 

3x² = 1

 

  x² = ±0.333

  x = -0.5777

  x = 0.5777

 

x = -0.5777       and        x = 0.5777