extended mean-value theorem

Let $f:[a,b]\to ℝ$ and $g:[a,b]\to ℝ$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists some number $\xi \in (a,b)$ satisfying:

$$(f(b)-f(a))g^{\prime }(\xi )=(g(b)-g(a))f^{\prime }(\xi ).$$

If $g$ is linear this becomes the usual mean-value theorem.