Exercise 7, § 4

Let $f$ be a convex function on the interval $]a,\ b[$ where $a \geq 0$; show that on this interval the function $x \mapsto f(x) - x f'(x)$ (the "ordinate at the origin" of the right semi-tangent at the point $x$ to the graph of $f$) is decreasing (strictly decreasing if $f$ is strictly convex).

Deduce that:

a) If $f$ admits a finite right limit at the point $a$ then $(x-a)f'_d(x)$ has a right limit equal to 0 at this point.

b) On $]a,\ b[$ either $f(x)/x$ is increasing, or $f(x)/x$ is decreasing, or else there exists a $c \in ]a,\ b[$ such that $f(x)/x$ is decreasing on $]a,\ c[$ and increasing on $]c,\ b[$.

c) Suppose that $b = +\infty$: show that if

is finite, then so is $\alpha = \lim_{x \to +\infty} f(x)/x$, and that the line $y = \alpha x + \beta$ is asymptotic $^5$ to the graph of $f$, and lies below this graph (strictly below if $f$ is strictly convex).