# Prove that there is a point $c\in(a,b)$ such that $f'(c)=0$.

Let $$f:[a,b]\rightarrow\mathbb{R}$$, $$0, a function that is differentiable and bijective such that $$\int_{f(a)}^{f(b)}f^{-1}(x)dx=0$$. Prove that there is a point $$c\in(a,b)$$ such that $$f'(c)=0$$. I changed the variable and $$\int_{f(a)}^{f(b)}f^{-1}(x)dx=\int_{a}^{b}f(f^{-1}(x))f'(x)dx=\int_{a}^{b}xf'(x)dx=0$$. So $$\int_{a}^{b}xf'(x)dx=0$$. From the mean value theorem, there is a $$c\in(a,b)$$ such that $$\int_{a}^{b}xf'(x)dx=f'(c)(b-a)$$ and from here I obtain the conclusion. Is this proof complete?