Let's say I have two functions $f(x$) and $g(x)$ and they are defined only for positive values of $x$. $f(x)$ is an increasing function of $x$, while $g(x)$ is a decreasing function of $x$. Both functions are probability distribution functions, so their values are between $0$ and $1$. Their product $f(x)g(x)$ will have some maximum and it will go to zero as $x$ increases. Because, $f(x)$ will go to $1$, while $g(x$) will go to $0$, thus their product will go to zero as $x$ goes to infinity.
I am trying to prove the following statement:
$ \max \left( f(a_1 x) g(x) \right) > \max \left( f(a_2 x) g(x) \right) $ for given $a_1$ and $a_2$ with $a_1 > a_2$, and $a_1$ and $a_2$ are positive numbers.
In other words, the maximum value of the product will increase with increasing argument of the first function $f(x)$. I couldn't prove this formally, although it seems quite intuitive. Any help will be much appreciated.