Let $f:[a,b]\rightarrow\mathbb{R}$, $0<a<b$, 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?


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