Maximum Principle: Suppose that f:(a,b)→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)≥f(x) for a<x<b. Let p be a subdifferential of f at c:f(x)≥f(c)+p(x−c) for a<x<b. Combining these inequalities yields f(x)≥f(x)+p(x−c) or 0≥p(x−c) for a<x<b. Hence p=0 hence f(x)≥f(c) from the subdifferential inequality. Thus f(x)=f(c) for a<x<b.
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