# How to find in which points both partial derivatives are equal to zero?

I need to find points in which both partial derivatives are equal to zero. Partial derivative with respect to $x$ is equal to: $(1-x)ye^{-x-y^2}$ With respect to $y$: $(1-2y^2)xe^{-x-y^2}$

When I set both of them equal to zero, I get from the first equation $y=0$, $x=1$ and from the second $x=0$ and $y=\pm 1/\sqrt 2$. But this seems that I am not on the right track.

Would be grateful for any tips. Thank you!

• The conditions you get from each partial derivative are alternatives, from the first, you get $y = 0$ or $x = 1$. – Daniel Fischer Jan 11 '14 at 11:14

## 1 Answer

Since $$(1-x)ye^{-x-y^2}=0\Rightarrow x=1\ \text{or}\ y=0,$$ $$(1-2y^2)xe^{-x-y^2}=0\Rightarrow x=0\ \text{or}\ y=1/\sqrt{2}\ \text{or}\ y=-1/\sqrt 2,$$ you'll have $$(1-x)ye^{-x-y^2}=0\ \text{and}\ (1-2y^2)xe^{-x-y^2}=0\Rightarrow (x,y)=(0,0),(1,\pm 1/\sqrt 2).$$

Edit : You have $2\times 3=6$ possibilities as the followings (Note that 'or' and 'and'):

$$x=1\ \text{and}\ x=0,$$ $$x=1\ \text{and}\ y=1/\sqrt 2,$$ $$x=1\ \text{and}\ y=-1/\sqrt 2,$$ $$y=0\ \text{and}\ x=0,$$ $$y=0\ \text{and}\ y=1/\sqrt 2,$$ $$y=0\ \text{and}\ y=-1/\sqrt 2.$$

You'll see some of these have to be excluded.

• Thank you very much for your answer! Could you be so kind to explain how you get from having roots of the equations to finding the points? – javaniewbie Jan 11 '14 at 11:31
• @javaniewbie: I added a bit. I hope this helps. – mathlove Jan 11 '14 at 11:36
• mathlove, you rock! I got it! Thank you very much! – javaniewbie Jan 11 '14 at 11:38
• You are welcome! – mathlove Jan 11 '14 at 11:38