I need to find a function $f(x)$ that maximizes a functional: $$ J(f)= \int\limits_{-\infty}^{+\infty} e^{-x^2/2}f(x) \,dx$$
Where $$f(x)>0 \ \text{ and} \int\limits_{-\infty}^{+\infty} f(x) \,dx = 1$$ Euler-Langrange equation will simply be:
$$ e^{-x^2/2}=0$$
So does it mean that there are no stationary points for this functional? But how do I search for a maxima/minima then?
Thank you!