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Finding Min/Max Saddle points of 2-var function

  1. Oct 15, 2008 #1
    1. The problem statement, all variables and given/known data
    Find local extrema and saddle points (if any) of the function [itex]f(x,y)=x^4+y^4-4xy-10[/itex]

    3. The attempt at a solution

    Taking the partials and setting them equal to zero I find:

    [itex]f_x=4x^3-4y=0\Rightarrow x^3-y=0[/itex] (1)
    [itex]f_y=4y^3-4x=0\Rightarrow y^3-x=0[/itex] (2)

    Solving (1) for y yields [itex]y=x^3[/itex]

    Plugging into (2) yields [itex](x^3)^3-x=0[/itex]

    [itex]\Rightarrow x(x^8-1)=0[/itex]

    Now what? I know that x=0 is one solution.

    Should I separate the expression in the brackets in to the difference of two squares?

    I am reluctant since that will yield [itex]x(x^4-1)(x^4+1)=0[/itex]
    [itex]\Rightarrow x(x^2-1)(x^2+1)(x^4+1)=0[/itex]
    [itex]\Rightarrow x(x-1)(x+1)(x^2+1(x^4+1)=0[/itex]

    and what the crap good is this?

    I know x={0,1,-1} and it looks like they just have a bunch of multiplicities.

    I guess that does work....so maybe I don't have any other question except:

    Is this the right wat to go about solving this?

  2. jcsd
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