# Thread: trouble finding critical points on a function, R^2 -> R

1. ## trouble finding critical points on a function, R^2 -> R

Q: Find the local maximum and minimum values and saddle point(s) of the function
$\displaystyle f(x, y) = (x^2 + y^2)e^{y^2 - x^2}$

I've found the the function's gradiant vector:

$\displaystyle \triangledown{f} = [-2xe^{y^2-x^2}(x^2+y^2-1), 2ye^{y^2-x^2}(x^2+y^2+1)]$

but I'm having trouble manipulating these equations to find where $\displaystyle \triangledown{f}=\vec{0}$. BOB says these points are $\displaystyle (0, 0)$ and $\displaystyle (\pm{1}, 0)$, but I'm not sure how to go about proving these are in fact the only points.

2. ## Re: trouble finding critical points on a function, R^2 -> R

An exponential is never 0 so you can divide $\displaystyle \nabla f= \vec{0}$ by $\displaystyle e^{y^2- x^2}$ to get $\displaystyle \nabla f= [-2x(x^2+ y^2- 1), 2y(x^2+ y^2+ 1]= \vec{0}$.

Since a vector is 0 if and only if each component is 0, we must have $\displaystyle -2x(x^2+ y^2- 1)= 0$ and $\displaystyle 2y(x^2+ y^2- 1)= 0$. And since ab= 0 if and only if either a= 0 or b= 0 we must have x=0, or $\displaystyle x^2+ y^2- 1= 0$ and y= 0 and $\displaystyle x^2+ y^2- 1= 0$. What are the solutions to those equations? ($\displaystyle (0, 0)$ and $\displaystyle (\pm1, 0)$ are solutions but there are two other solutions also.)