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

Decide whether F=x^{2}y^{2}-2x-2y has a minimum at the point x=y=1 (after showing that the first derivatives are zero at that point).

2. Relevant equations

F_{xx}F_{yy}-F_{xy}^{2}

3. The attempt at a solution

So I found that:

F_{x}=2xy^{2}-2, which at point (1,1) = 0 OK

F_{y}=2x^{2}y-2, which at point (1,1) = 0 OK

F_{xx}=2y^{2}

F_{yy}=2x^{2}

F_{xy}=4xy

So at point (1,1):

F_{xx}F_{yy}-F_{xy}^{2}=4-16=-12, which is less than 0.

Does this mean that because it is less than zero, rather than greater, it DOES NOT have a minimum but rather a saddle point? Thanks!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Minimum Point of a Function

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