Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 1 / 11E (Analysis I)

Differentiability at $x$ means that the limitexists as a finite real number.

If $f$ is continuous on $[x,y]$ and differentiable on $(x,y)$, then some $c\in (x,y)$ satisfies $f(y)-f(x)=f\prime (c)(y-x)$.

If $f\prime \ge 0$, the mean value theorem gives $f(y)-f(x)=f\prime (c)(y-x)\ge 0$. Conversely, if $f$ is increasing, every difference quotient with positive or negative increment is nonnegative; taking its limit gives $f\prime (x)\ge 0$.

For $x>0$, apply the mean value theorem to $[0,x]$: $f(x)/x=f\prime (c)$ for some $c<x$. Since $f\prime$ is increasing, $f(x)/x\le f\prime (x)$. Thereforeand part (c) proves that the quotient is increasing.