Thread: differentiability of function g

Suppose g is a function continuous on the closed and bounded interval
[a, b], b > a > 0 and differentiable on the open interval (a, b).
Prove that there exists a point β in (a, b) such that
.

I tried to use the Mean Value Theorem, but I could not arrive at the answer.

Think about the mean value of g(x) and the mean value of g'(x). Remember, g'(x) has to, at one point, be equal to the average rate of change g(x)