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?.


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