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 March 18th, 2012, 05:04 PM #1 Senior Member   Joined: Jan 2010 Posts: 324 Thanks: 0 Finding extrema points and values So we have a function F(x,y) such that x^3+3xy+y^3 on the domain formed by -2<=x<=2 and -2<=y<=2 he wants us to find the criticle points and have the coordinates and function values.. wanna help me out with this?
March 18th, 2012, 06:14 PM   #2
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Re: Finding extreema points and values

The first partial derivatives are:

$F_x(x,y)=3x^2+3y$ and $F_y(x,y)=3x+3y^2$

Equating both to zero implies that we have the critical points:

(0,0), (-1,-1)

To determine whether extrema exist at these points, we may employ the second partials test:

Let $D(x,y)=F_{xx}(x,y)F_{yy}(x,y)-$$F_{xy}(x,y)$$^2=(6x)(6y)-$$3$$^2$

First point:

$D(0,0)=-9<0=$

This point is not an extremum.

Second point:

$D(-1,-1)=36-9=27>0=$

$F_{xx}(-1,-1)=6(-1)=-6<0=$

This point is a relative maximum. The local maximum has a value of:

$F(-1,-1)=1$

Here is a 3D plot with contour lines:

[attachment=0:1a7k5mr3]3dplot.jpg[/attachment:1a7k5mr3]
Attached Images
 3dplot.jpg (29.4 KB, 77 views)

 March 20th, 2012, 08:34 PM #3 Senior Member   Joined: Jan 2010 Posts: 324 Thanks: 0 Re: Finding extrema points and values He gave us a list of ten points though. like (0,0) .... (-sqrt 2, -2)...etc.

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