# Convexity of product of two given functions

Let $f(x)$ be a convex function of $x$ in a given positive interval. Also assume $f(x)\geq 0$ everywhere in that interval. Is the function $g(x)=xf(x)$ convex in that interval?. Is it possible that $g(x)$ is monotonically increasing or monotonically decreasing.

• To address the last part of your Question, can $g(x)$ be monotone, yes, this is possible for $g(x) = xf(x)$ of the form you specify. For example, if $f(x)= x^2$ then $g(x) = x^3$ is monotone increasing on every (positive) interval. But perhaps you have another "possibility" in mind. – hardmath Jul 14 '14 at 13:30

N0. Let $f\colon(0,\infty)\to(0,\infty)$ be defined by $f(x)=x^{-p}$, $0<p<1$. $f$ is convex, since $$f''(x)=-p(p-1)x^{p-2}>0\quad\forall x>0.$$ Howevwe $$x\,f(x)=x^{1-p}$$ is concave.
• does this mean that nothing can be said about the nature of $g(x)$ in general ? – dineshdileep Jan 28 '14 at 17:22
Let $f:[0,1]\to\mathbb R$ be defined by $f(x)=1-x$. This function is convex. But, $g(x)=x-x^2$, which is not convex. To get a monotonically increasing $g$, take $f(x)=1$. To get a monotonically decreasing $g$, take $f:[1,2]\to\mathbb R$, defined by $f(x)=\frac{2-x}x$.