Finding the critical points of this multivariable function

Question

I have just been practising finding critical points for multivariable functions and have been working on this question

Find the critical points of $f(x,y)=x^2e^y$

so far I have found the gradient vector of this function

$\bigtriangledown f = (2xe^y, x^2e^y)$

and I know the x coordinate of the critical points would be $x= 0$ but I cant seem to find a y coordinate. So if I cant find a y coordinate I cannot go on to then classify the points

Any help would be much appreciated

Answer 1

Critical points have $x=0$, as you have found. You have also found that it doesn't matter what $y$ value you plug in, it always leads to $\nabla f = 0$. This means that all critical points are of the form $(0, y)$ for any $y$. You should be able to do the usual classification of min, max, or saddle, using $y$ as a variable.