How to find where the magnitude of the gradient of a function is maximized?

Question

How to find where the magnitude of the gradient of ${{e}^{-({{x}^{2}}+{{y}^{2}})}}$ maximizes? I managed to calculate the magnitude - $2{{e}^{-({{x}^{2}}+{{y}^{2}})}}\sqrt{{{x}^{2}}+{{y}^{2}}}$

Answer 1

The magnitude of the gradient just depends on the distance from the origin $\rho=\sqrt{x^2+y^2}$.

The function $f:[0,+\infty]\to[0,+\infty]$ defined by: $$ f(\rho) = 2\rho\, e^{-\rho^2} $$ attains its maximum when $f'=0$. Since: $$ f'(\rho) = 2(1-2\rho^2)e^{-\rho^2} $$ the maximum is attained for $\rho=\frac{1}{\sqrt{2}}$: $$ f(\rho)\leq \sqrt{\frac{2}{e}}.$$