Exercise 15, § 1

Let $f$ be a numerical function continuous, strictly increasing in an interval $[0, a]$, and such that $f(0) = 0$; let $g$ be its reciprocal function, defined and strictly increasing in $[0, f(a)]$; now prove that, for $0 \leq x \leq a$ and $0 \leq y \leq f(a)$, one has

the equality holding only if $y = f(x)$ (study the variations as a function of $x$ (y remaining fixed) of $xy - \int_0^x f(t) \, dt$). Deduce that, for $x \geq 0, y \geq 0, p > 1, p' = p/(p - 1)$, one has $xy \leq ax^p + by^{p'}$ for $a > 0, b > 0$ and $(pa)^p (pb')^{p'} \geq 1$.