Exercise 15, § 1

Let $\mathbf{f}$ be a continuous real function, strictly increasing on an interval $[0, a]$, and such that $f(0) = 0$; let $g$ be its inverse function, defined and strictly increasing on $[0, f(a)]$, show that

for $0 \leq x \leq a$ and $0 \leq y \leq f(a)$, equality occurring only if $y = f(x)$ (study the variation of $xy - \int_0^x f(t) \, dt$) as a function of $x$ ($y$ remaining fixed). 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 $(pa)^{p'} (p'b)^{p'} \geq 1$.