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Let $f:[a,b]\rightarrow \mathbb R$ be a continuous function such that

$$\frac{f(x) - f(a)}{x-a}$$

is an increasing function of $x\in [a,b]$. Is $f$ necessarily convex? What if we also assume that

$$\frac{f(b) - f(x)}{b-x}$$

is increasing in $x$?

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    I am adding the condition that $f$ be continuous.2012-07-18

2 Answers 2