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Critical Point(s) of a Multivariable Function

  1. Mar 25, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the critical points of f.
    [tex]f(x,y)=2+\sqrt{3(x-1)^2+4(y+1)^2}[/tex]


    2. Relevant equations
    For fx(x,y) I get:
    [tex]f_x(x,y)=0+\frac{1*6(x-1)}{2\sqrt{3(x-1)^2+4(y+1)^2}}=\frac{3(x-1)}{\sqrt{3(x-1)^2+4(y+1)^2}}[/tex]
    For fy(x,y) I get:
    [tex]f_y(x,y)=0+\frac{1*8(y+1)}{2\sqrt{3(x-1)^2+4(y+1)^2}}=\frac{4(y+1)}{\sqrt{3(x-1)^2+4(y+1)^2}}[/tex]


    3. The attempt at a solution
    Solving both of these for x and y when set equal to 0, gets me x = 1 and y = -1. However neither of these functions exist when x and y equal those values. Does the original function still have a critical point at (x,y) = (1,-1)?

    Additionally, when I put the function into Wolfram Alpha it says that it has no critical points.
     
    Last edited: Mar 25, 2013
  2. jcsd
  3. Mar 26, 2013 #2
    What is the definition of the critical point?
     
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