Is exponential of a concave function concave?

Question

is this function:

$$\exp\Big(-||Ax||^2\Big)$$

concave in A??

I know that exponential of a convex function is convex, but is exponential of a concave function concave??

Answer 1

Did's example

Let $f(x)=e^{-x^2}$.

Then $f'(x)=-2xe^{-x^2}$, so $f''(x)=4x^2e^{-x^2}-2e^{-x^2}=(4x^2-2)e^{-x^2}$.

$e^{-x^2}$ is always positive, but $4x^2-2$ can be positive or negative.