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Suppose I have a function $f(x):[a,c]\rightarrow \mathbb{R}$ defined as $f(x)=\cases{f_1(x),\ a\leq x\leq b\\f_2(x),\ b<x\leq c}$ where $f_1(x),f_2(x)$ are two concave functions over the domains $[a,b],\ [b,c]$ respectively. Given that $f_1(b)=f_2(b)$ and that $f_1'(b)=f_2'(b)$ (where the derivatives at $b$ are naturally left for $f_1$ and right for $f_2$), how do I prove that $f(x)$ is concave? thanks a lot!

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Here is a straightforward proof if $f_k$ are differentiable:

A differentiable $f$ is concave iff $f'$ is non increasing.

Since both $f_k$ are non increasing, and $f'_1(b) = f'_2(b)$, we see that $f$ is non increasing, hence concave.

This can be generalised in a number of ways, but the proofs a little messier.

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  • $\begingroup$ Thanks. I was missing the lemma that says that > A differentiable $f$ is concave iff $f′$ is non increasing. do you have a reference that proves it? $\endgroup$ – avishay antman Dec 5 '16 at 22:53
  • $\begingroup$ I don't have access to my library at the moment, but you could look at en.wikipedia.org/wiki/Convex_function#Properties, Properties #1 & #4. $\endgroup$ – copper.hat Dec 5 '16 at 23:34

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