my problem is how to find the maximum of an exponential sum. The interested function is $f(x)=\sum^M_{m=1}\exp(a_mx^2+b_mx+c_m)$. $a_m$, $b_m$ and $c_m$ for all $m$ are known coefficients and $M$ is known as well.
However, finding the maximum of $f(x)$ seems not straightforward as it is not a monotone function. If trying to find the roots of $\text{d}f(x)/\text{d}x$ and obtain the extrema, it seems still can not be done.
My anther idea is to use exponential series to approximate $f(x)$, like $e^x=\sum^\infty_{n=0}\frac{x^n}{n!}$, and do the derivative. However, the maximum or extrema still not can be found.
Does anyone has suggestion to solve this problem? By using approximation, semi-analytical method or some else?
Thank you.