Find the critical points of this function:

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  • #1
knowlewj01
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Homework Statement



Find the critical points of this function and determine wether they are local maxima, minima or saddle points...

[itex]f=\frac{1}{x} + \frac{1}{y} + xy[/itex]


The Attempt at a Solution



start off by partially differentiating and setting to zero for x and y:

[itex] \frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0[/itex]


[itex]\frac{\partial f}{\partial x}= -\frac{1}{x^2} + y = 0[/itex]

[itex]\therefore y=\frac{1}{x^2}[/itex]


[itex]\frac{\partial f}{\partial y}= -\frac{1}{y^2} + x = 0[/itex]

[itex]\therefore x=\frac{1}{y^2}[/itex]

so:

[itex]\exists[/itex] a stationary point at [itex](\frac{1}{y^2},\frac{1}{x^2})[/itex]

using [itex] D = f_{xx}f_{yy} - (f_{xy})^2[/itex]




Does all of this look ok so far? because i do not get a definitive answer for where the critical point lies or any indication of it's nature.
 

Answers and Replies

  • #2
Fightfish
954
117
No, both conditions have to be satisfied. Surely you can't leave your coordinates in terms of each other - they can and should then be simplified! So, it is a matter of solving the equations
[tex]y=\frac{1}{x^2}[/tex]
[tex]x=\frac{1}{y^2}[/tex]
simultaneously.
 

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