Slope inequalities for convex functions

Lemma. Let $f:I\to\mathbb{R}$ be a convex function on a nonempty interval $I\subset\mathbb{R}$. For any $a,b\in I$ with $a<b$ and any $x\in(a,b)$,

Context. Convexity forces secant slopes to increase as you move to the right. This lemma is the key step in relating convexity to monotonicity of derivatives.

Proof sketch. Write $x=tb+(1-t)a$ with $t=(x-a)/(b-a)\in(0,1)$ and apply Jensen’s inequality to compare $f(x)$ with the linear interpolation between $f(a)$ and $f(b)$; rearranging yields the slope bounds.