Maximum Principle: Suppose that $f:(a, b) \rightarrow \mathbf{R}$ is convex and has a maximum at some $c$ in $(a, b)$. Then, $f$ is constant (use subdifferentials).
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If $c$ is a maximum of $f$, then $f(c) \geq f(x)$ for $a<x<b$. Let $p$ be a subdifferential of $f$ at $c: f(x) \geq f(c)+p(x-c)$ for $a<x<b$. Combining these inequalities yields $f(x) \geq f(x)+p(x-c)$ or $0 \geq p(x-c)$ for $a<x<b .$ Hence $p=0$ hence $f(x) \geq f(c)$ from the subdifferential inequality. Thus $f(x)=f(c)$ for $a<x<b$.
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