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Finding critical points

  1. Nov 13, 2015 #1
    1. The problem statement, all variables and given/known data
    find all critical points and identify the locations of local maximums, minimums and saddle points of the function [tex]f(x,y)=4xy-\frac{x^4}{2}-y^2[/tex]

    2. Relevant equations


    3. The attempt at a solution
    setting Partial derivative respect to x = 0 : [tex]4y-\frac{4x^3}{2}=0[/tex]
    setting partial derivative respect to y=0: [tex]4x-2y=0[/tex]

    from the second equation, Y=2X, plug it into the first equation. I get : [tex]2x(4-x^2)=0[/tex]
    now, x=0 or x=-2 or x=2. since y=2x, now I got 3 sets of critical points. (0,0), (-2,-4) and (2,4)

    second partial derivative respect to x = [tex]-6x^2[/tex]
    second partial derivative respect to y=-2
    and ƒxy=0

    so [tex]D(a,b)= 12x^2[/tex]

    plug in these critical points into D(a,b) I found that local max value at (-2,-4) and (2,4) for which both give D value of 48, greater than 0, and ƒxx at both points are smaller than 0.
    for D(0,0) , the test is inconclusive .

    I did not find any local min, I am wondering if i missed something.
    any help is appreciated
     
  2. jcsd
  3. Nov 13, 2015 #2

    ehild

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    fxy is not zero.
     
  4. Nov 13, 2015 #3

    Ray Vickson

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    ##f_{xy} \neq 0##.
     
  5. Nov 13, 2015 #4
    opps, got it. thank you
     
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