Hi everyone. I have a question on one of my homework problems.

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

So I started doing this problem and it was looking really messy so I went on cramster and got this:

Step 1

Step 2

There's already a mistake here since you forgot to differentiate the second summand above. It should be:

$\displaystyle f_x=2xe^{y^2-x^2}-2x^3e^{y^2-x^2}-2xy^2e^{y^2-x^2}=0\Longrightarrow 2xe^{y^2-x^2}(1-x^2-y^2)=0$ , and this

alone already tells you that all the points on the unit circle around the origin are zeroes of this derivative...

Likewise $\displaystyle f_y$ is wrong...you seem to have some problems with partial derivatives. Check this.

Tonio
Step 3

so critical points are (0,0), (1,0), and (-1,0)

Step 4

Step 5

at (0,0):

at (1,0):

at (-1,0):

Therefore, f has a local minimum of 0 at (0,0) and saddle points at (1,0) and (-1,0).

The answer is correct but the work makes no sense. When computing $\displaystyle f_x$, $\displaystyle y^2e^{y^2-x^2}$ was completely neglected. $\displaystyle f_x$ should have been:

$\displaystyle f_x = 2xe^{y^2-x^2}-2x^3e^{y^2-x^2} -2xy^2e^{y^2-x^2} = 0$

Same problem with $\displaystyle f_y$

Is there something wrong with the solution or am I doing something wrong?

Thanks in advance!