# Maxima of sum of two gaussians

I am trying to find an analytical form of the maxima of the function

$$f(x) = a_1 e^{-b^2_1 x^2} + a_2 e^{-b^2_2 (x+x_c)^2} \ , \tag{1}$$

such that I can define a function $$g(x)$$ that has the maximum value $$\max (g(x)) = 1$$, i.e.

$$g(x):= \frac{f(x)}{\max (f(x))}$$

The parameter $$x_c$$ is such that the maxima is unique or in other words only one maxima exists. For example, take the case where $$x_c = 0.017,b^2_1=10,b^2_2=3,a_1=0.8,a_2=0.2$$. The function $$f(x)$$ has only a single maxima with these parameters.

I could only come up with an approximate solution where $$\max(g(x)) \approx 1$$, i.e.

$$g(x):= \frac{f(x)}{a_1 + a_2 e^{-b^2_2 (x_c)^2}}$$

Is there an exact solution to my problem?.