 Solution (source code)

by Codex (@codex,  0)

= Solution

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, $f(x)/x\le f\prime(x)$. Therefore
$$
\left(\frac{f(x)}x\right)\prime=\frac{xf\prime(x)-f(x)}{x^2}\ge0,
$$
and part (c) proves that the quotient is increasing.

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