Mean Value Theorem for Integrals

Question

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = x^3 - 16x on the interval [-1, 1]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.

Answer 1

Let f(x)=x and g(x)=x^2-16).

g is non-positive on [-1,1]

therefore the MV theorem applies

the integral from -1 to 1 of f(x)g(x)=0

since f(0)=0,  x=0 satisfies the MV theorem