Given that we have a function $f(x)$ which is a sum of two arbitrary Gaussian functions such that: $$ f(x) = A_{1} e^{-\frac{(x - \mu_{1})^{2}}{2\sigma_{1}^{2}}} + A_{2} e^{-\frac{(x - \mu_{2})^{2}}{2\sigma_{2}^{2}}} $$ where weights $A_{1}$ and $A_{2}$ are positive, and $\sigma_1$ and $\sigma_2$ denote the standard deviations of the Gaussian functions.
I have found out that there is no analytical solution in obtaining the local maxima of the function $f(x)$. However, I was wondering if anyone is aware of any way to approximate this problem analytically (and not numerically).
I basically need this for modelling a specific problem and I would like to see if there are any cool tricks or any theorems that may help me to do some rough approximations to simplify this problem.
Thanks, Charis