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

$\displaystyle \frac{bg(a)-ag(b)}{b-a} = g(\beta) - \beta{g'(\beta)$.

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