Max/Min/Saddle Points of an Equation

CalcGeek31 · #1 $Old$ November 4th, 2009, 07:06 PM

Find the relative extrema and saddle points of the function f(x, y) = 4xy – x4 – y4. Include the second derivative test in your answer.

I know how to get the partials, for x, y, xx, yy, and xy, but after that, I am lost on this type of question, any help is appreciated.

Scott H · #2 $Old$ November 4th, 2009, 07:48 PM

For problems involving maxima and minima in multiple variables, we calculate the discriminant

$D=f_{xx}f_{yy}-f_{xy}^2$

at critical points. If $D>0$ and one of the partial derivatives is nonzero, then the critical point is a relative extremum, if $D<0$ , the critical point is a saddle point, and if $D=0$ , the test is inconclusive.

HallsofIvy · #3 $Old$ November 5th, 2009, 05:27 AM

Quote:

Originally Posted by Scott H $View Post$

For problems involving maxima and minima in multiple variables, we calculate the discriminant

$D=f_{xx}f_{yy}-f_{xy}^2$

at critical points. If $D>0$ and one of the partial derivatives is nonzero, then the critical point is a relative extremum, if $D<0$ , the critical point is a saddle point, and if $D=0$ , the test is inconclusive.

Strictly speaking that only works for functions of two variables (which this problem is, of course). The conditions for max, min, saddle point, for functions of more than two variables are, unfortunately, much more complicated.

Scott H · #4 $Old$ November 6th, 2009, 05:45 PM

Yes, thank you for the correction.