(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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Minimum Point of a Function

**Physics Forums | Science Articles, Homework Help, Discussion**