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 $x$?
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 $x$?