Given a function $h$: $ h(x)=af(x)−b[1−F(x)], $ where $a$ and $b$ are constants with $b>0$, $f$ is a probability (a generalized) density function and $F$ is its CDF, I want to prove that there exists an $x$ such that $h(x)=0$.
I was trying to use the Extreme value theorem, but I got it difficult to find the limit of $f(x)$ as $x$ approaches $±∞$. Taking $\lim f(x)=0$ as $x→±∞$, the result I found is $h(∞)=0$ and $h(-∞)=0$, does the envelope theorem apply in this case? Please suggest me some ways to prove the claim.