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Finding the Critical Point(negative square root)

  1. Nov 5, 2014 #1
    1. The problem statement, all variables and given/known data
    Find the critical points for the function g(x, y, z) = x3+xy2+x2+y2+3z2.

    2. Relevant equations


    3. The attempt at a solution
    I've come up with the following 3 equations (derivatives set so that they are equal to 0)
    (1) 3x2+y2+2x=0
    (2) 2xy+2y=0
    (3) 6z=0

    From (3),
    z=0​
    From (2),
    2y(x)+2y(1)=0
    x+1=0
    x=-1​
    From (1) using what I have obtained from (2),
    3(-1)2+y2+2(-1)=0
    3-2+y2=0
    1=-y2
    y2=-1​
    I do not know how I am supposed to isolate y in this case since square roots are supposed to be positive. Do I need to get i involved? If so, how?
     
  2. jcsd
  3. Nov 5, 2014 #2

    pasmith

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    Homework Helper

    From (2) you have two possibilities: [itex]0 = 2xy + 2y = 2y(x + 1)[/itex] so either [itex]y = 0[/itex] or [itex]x = -1[/itex] or both.

    You must also satisfy (1). You've shown that if [itex]x = -1[/itex] then (1) requires that [itex]y^2 = -1[/itex], so that doesn't give you a critical point. There remains the [itex]y = 0[/itex] possibility.
     
  4. Nov 5, 2014 #3

    Ray Vickson

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    ##(2) \Longrightarrow 2y(x+1) = 0 \Longrightarrow y = 0 \; \text{or }\; x+1 = 0##.
     
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