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Find the critical points of this function:

  1. Mar 24, 2010 #1
    1. The problem statement, all variables and given/known data

    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]


    3. 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.
     
  2. jcsd
  3. Mar 24, 2010 #2
    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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