For positive constants $a,b,c,d$, I'm interested in an upper bound for $g(x)=(ax+b)e^{-cx^2-dx}$ for $x\geq 0$ in terms of the constants.
One approach is just to solve the maximization problem $\max_{x\geq 0}g(x)$, but it cannot be done in closed form (I think).
A simple approach is to bound $ax+b\leq e^{ax+b-1}$, which yields an upper bound for $g$ that can be maximized. I just need one "good" bound for $g(x)$ and this feels like an studied problem, any reference will be much appreciated.
Thanks!
You can apply the classic method of finding points where the derivative is 0. Those are going to be local extrema of your function. Then, a quick study of the sign of the derivative will lead you to finding the exact global maximum.
Take your function $g : x \mapsto (ax + b)e^{-cx^2 - dx}$ and derivate it. It gives :
$g'(x) = (a + (ax + b)(-2cx - d))e^{-cx^2 - dx} \\ = (a - bd - 2bcx - adx - 2acx^2)e^{-cx^2 - dx}$
Can you take it from there ?