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-   -   Extremum (http://mymathforum.com/calculus/4953-extremum.html)

 StevenMx December 16th, 2008 07:39 AM

Extremum

I've got another interesting (I hope) problem and I hope you'll help me to solve it :) Well it goes like this: Determine all points, where function f has it's extremums, and define whether it is maximum or minimum, if:

$f(x,y)=x^{2} - xy + y^{2} - 4ln|x| - 10ln|y|$

 Scott December 16th, 2008 09:19 AM

Re: Extremum

There is a theorem for twice-differentiable functions f(x, y) that if

$1. \,f_x\,=\,f_y\,=\,0$ at $(x,\,y)$

$2.\,f_{xx}f_{yy}\,-\,f_{xy}^2\,>\,0$ at $(x,\,y)$

$3.\,f_{xx}\,<\,0$ at $(x,\,y)$

then f(x, y) has a relative maximum at (x, y), and similarly a relative minimum if $f_{xx}\,>\,0$. (I believe it can be proved with Taylor's theorem.)

 December 16th, 2008 09:25 AM

Re: Extremum

$f_{x}=2x-\frac{4}{x}-y$...[1]

$f_{y}=2y-\frac{10}{y}-x$...[2]

Set each to 0. Solve [1] for y and sub into [2]. Then, solve for x.

You can get y by subbing the x's back into the y you found. Check the critical points by using the second partials derivative test. It can be found in any calc book.

$D=f_{xx}f(x,y)f_{yy}-f_{xy}^{2}(x,y)$

 StevenMx December 16th, 2008 10:13 AM

Re: Extremum

Thank you :) You are really helpful guys :)

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