Finding the local minima,maxima and saddle point of $f(x,y)=y^2e^{-x}-x^2$

Question

I did this question and got one critical point $(0,0)$. Whilst solving it using Hessian Matrix method, the $D(0,0)$ was found as $0$ which makes it inconclusive. However,I am having doubts on the way I did it. So help would be really appreciated.

Answer 1

Indeed the only critical point is (0,0). Then using D=$f_{xx}$$f_{yy}$-($f_{xy}$)^2 at the critical point (a,b)=(0,0), we see that D<0, which means that the critical point is a saddle point. http://tutorial.math.lamar.edu/Classes/CalcIII/RelativeExtrema.aspx https://www.wolframalpha.com/input/?i=f%28x%2Cy%29%3Dy%5E2e%5E%28-x%29%E2%88%92x%5E2