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Critical points of 2-Variable Function

  1. Oct 7, 2008 #1
    1. The problem statement, all variables and given/known data
    Find local max/min, and saddle points (if any) of

    [tex]f(x,y)=x^2+y^2+x^2y+4[/tex]

    This should be simple, but I am having algebra-block on solving the partial derivatives to find the critical points.

    [itex]f_x=2x+2xy=0[/itex] (1)
    [itex]f_y=2y+x^2=0[/itex] (2)

    If I multiply the second equations by -x an add it to the second and solve for x, I get

    x={0,+sqrt2, -sqrt2}

    But for some reason I cannot figure out how to solve equation 2 ?

    Why am I retarded?
     
  2. jcsd
  3. Oct 7, 2008 #2

    gabbagabbahey

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    Well, for x=0 equation 2 gives: [itex]2y+(0)^2=0 \Rightarrow y=0[/itex] How about for x=+sqrt2?
     
  4. Oct 7, 2008 #3
    I guess I don't understand this.

    I want to know at what points the slope of the tangent lines is 0.

    Now, I need values of x and y that satisfy both equations simultaneously. I am used to solving the equations simultaneously, not by assigning specific values to x or y.

    I am not sure why that bothers me so much. But as you say:

    if x=0 then eq 2 is satisfied by y=0, thus (0,0) is critical

    if x=+ or - sqrt2 eq 2 is satisfied by y=-1, thus (+sqrt2, -1) and (-sqrt2, -1) are critical, yes?
     
  5. Oct 7, 2008 #4

    gabbagabbahey

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    Yes, you can double check that those points satisfy both equations by substituting each point into them.
     
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