(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the maximum and minimum values of f(x,y) = (xy)^{2}on the domain x^{2}+ y^{2}< 2. Be sure to indicate which is which

2. Relevant equations

I am not sure what to put here. I solved this problem a different way, and I am not confident I did it correctly.

3. The attempt at a solution

f_{x}= 2xy^{2}= 0

f_{y}= 2x^{2}y = 0

only critical point is (0,0) and the value of f there is f(0,0)= 0. Therefore the local and absolute minimum value is 0.

I tested to make sure by estimating what the graph would look like, I have a lot of trouble graphing the traces once I do this on the three coordinate axes. It has hyperbolas in the first and third quadrants as xy traces. It has parabolas becoming ever steeper for the yz and xz traces with the vertex of them all being at (0,0) which proves that that the minimim is at the origin. I never took the restricted domain into account however because I thought it was irrelevant. Which makes me think I did something wrong. The restricted domain is a circle of radius 2.

Hope you can understand what I did, and thanks for your help!

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# Maxima and Minima on surfaces in three dimensional space

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