# Find the critical points of the function $f(x,y)=(x^2+y^2)e^{y^2-x^2}$

Find the critical points of the function $f(x,y)=(x^2+y^2)e^{y^2-x^2}$.

Now I have found the partial derivative of f(x,y) with respect to both x and y and got the following:

$\frac{\partial{f}}{\partial{x}}=-2xe^{y^2-x^2}(x^2+y^2-1)$

$\frac{\partial{f}}{\partial{y}}=2ye^{y^2-x^2}(x^2+y^2+1)$

Now I know for critical points, both the partial derivatives have to equal 0 but how do i go about finding the coordination of the points?

Thanks a million for any help!

• You should check $\partial f / \partial y$ again. – StackTD Apr 13 '16 at 11:47
• Oh ok thank you I think I can do it now! – user2250537 Apr 13 '16 at 11:52
• That looks better. If you're still stuck, shout! – StackTD Apr 13 '16 at 11:58
• Ok thanks, but no I think I got it, I got critical points at (0,0), (0, i), (0, -i), (1, 0) and (-1, 0) where i is the imaginary number. – user2250537 Apr 13 '16 at 12:02
• That's right, although I assume you're studying $f$ as a real-valued function? – StackTD Apr 13 '16 at 12:46