Hello,

Given two function $\displaystyle f:\mathbb{R}\to\mathbb{R}$ which is monotonic increasing and $\displaystyle g:\mathbb{R}\to\mathbb{R}$ which is monotonic decreasing I need to prove that $\displaystyle h(x)=f(x)g(x)$ is not monotonic.

I started my proof with assuming that $\displaystyle h(x)$ is monotonic increasing so for every $\displaystyle x_1<x_2$ I know that $\displaystyle h(x_1)<h(x_2)$ which means that $\displaystyle f(x_1)g(x_1)<f(x_2)g(x_2)$ but I'm stuck here and don't how to continue. I know that $\displaystyle f(x_1)<f(x_2)$ and $\displaystyle g(x_1)>g(x_2)$ but I don't know how to connect between them.

Thank you for your help.