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Let $\mathsf{f(x)}$ be a differential and an invertible function, such that $\mathsf{f''(x)>0}$ and $\mathsf{f'(x)>0}$.

Prove that$$\mathsf{ f^{-1}\left(\frac{x_1 + x_2 +x_3}{3} \right) > \frac{f^{-1}(x_1)+f^{-1}(x_2)+f^{-1}(x_3)}{3}}$$

I have no clue, how to start it. I think a graphical solution can be obtained but I am confused about the graph of the inverse function.

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    Hint: Did you learn something about the relationship between the sign of the second derivative of a function and convexity? If yes, can you calculate the second derivative of $f^{-1}$ in terms of $f, f^'$ and $f{''}$ ? Can you determine the sign of it?2012-01-02

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