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.

Not quite sure how to do this one at all to be completely honest. Thank you for helping!

1 answer

https://www.khanacademy.org/math/differential-calculus/derivative-applications/mean_value_theorem/v/mean-value-theorem-1

the average slope is (f(1)-f(=1))/(2)=-16/2=-8 check that.

then the points are
f'=-8=3x^2-9
x^2=1/3
x=+- sqrt (1/3)

check: f'(sqrt1/3)=1-9=-8
f'(-sqrt1/3)=1-9=-8