prove on differntiable function

Question

Let $f(x)$ be differentiable function $(a,b)$ and continuous on $[a,b]$ I need to prove that there exist $c \in (a,b) $ such that: $$ \frac{a\cdot f(a) -b \cdot f(b)}{a-b} = f(c) + c\cdot f'(c) $$

I'm not sure how to start... from Lagrange's theorem we can say that there exist $a < c < b$ such that:

$$ f'(c) = \frac{f(b) - f(a)}{b-a}$$

But Im not sure how to proceed..

Thanks

Answer 1

Just use Lagrange's theorem for $g(x):=x\cdot f(x)$.