Help with finding relative extrema and saddle points

[mathy555]

Hello everyone,

I have been big time stuck on a problem for several days now and have tried numerous resources for help but I just get further and further lost. The problem I'm dealing with is attached below.

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Examine the following function:

. . . . .\(\displaystyle f(x,\, y)\, =\, \left(-ax^2\, +\, by^2\right)^{-x^2\, -\, y^2}, \mbox{ with }\, a\, =\, -1\, \mbox{ and }\, b\, =\, 2\)

for relative extrema and saddle points. Make a plot of the function that shows all extrema and saddle points.

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Now I understand the steps for finding these...1) find both partial derivatives 2) set them to 0 and then you use rules and such and second derivative test.

Well I'm having big trouble with the partial derivatives and have a few different answers and setting them to 0 I'm stumped b/c I'm not good with multi-variables

If someone can help that would be great. I'm not just looking for an answer as I want to understand this

 

[ksdhart2]

The steps you've outlined are vague, but they appear to me to be the correct ones. However, we cannot provide any feedback as to where you may or may not be going wrong in applying these steps, as you have not shown any of your work. Please include any and all attempts at a solution, even if you know they're wrong. That should give us an idea of how best to proceed. Thank you.

 

[mathy555]

The steps you've outlined are vague, but they appear to me to be the correct ones. However, we cannot provide any feedback as to where you may or may not be going wrong in applying these steps, as you have not shown any of your work. Please include any and all attempts at a solution, even if you know they're wrong. That should give us an idea of how best to proceed. Thank you.

Thanks for the response. As for finding the partial derivatives I've come up with a few different answers. We are allowed to use Maple to do these problems if we know how and I used Maple as well (I don't have that answer on me b/c my computer doesn't have maple). What I'm also confused with is do you substitute the values for a and b into the equation before you find the partial derivative of it or no? Regardless the partial derivatives though seem quite long and given how they have multi variables I don't have much experience when finding the critical points of those functions.

My friend and I have tried to use some of the sites which are really helpful (such as wolframalpha and symbolab) along with Maple but we've gotten different results for the partial derivatives.

In my original post I was just stating I understand the process that needs to be done to find extrema and saddle points, however, when it comes to getting the correct partial derivatives (fx, fy, fxx, fxy) and then finding the critical points when set to 0 we are really stuck.