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Homework Help: Finding the critical points of a multivariable function and determining local extrema

  1. Dec 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Let f(x,y)=Ax2+E where A and E are constants. What are the critical points of f(x,y)? Determine whether the critical points are local maxima, local minema, or saddle points.


    2. The attempt at a solution

    First I found the first partial derivatives with respect to x and y
    [tex]\partial[/tex]f/[tex]\partial[/tex]x=2Ax
    [tex]\partial[/tex]f/[tex]\partial[/tex]y=0
    [tex]\Rightarrow[/tex] 2Ax=0,
    [tex]\Rightarrow[/tex] x=0 for any constant A.

    Therefore, all points lying on the y-axis are critical points.
    (i.e. C.P.'s = (0,n), n[tex]\in[/tex]R.)

    Now, we have to find the second partial's with respect to x and y.
    [tex]\partial[/tex]2f/[tex]\partial[/tex]x2=2A
    [tex]\partial[/tex]2f/[tex]\partial[/tex]y2=0
    and
    [tex]\partial[/tex]2f/[tex]\partial[/tex]x[tex]\partial[/tex]y=0

    Therefore Df=([tex]\partial[/tex]2f/[tex]\partial[/tex]x2)([tex]\partial[/tex]2f/[tex]\partial[/tex]y2)-([tex]\partial[/tex]2f/[tex]\partial[/tex]x[tex]\partial[/tex]y)2 at (0,n) , n[tex]\in[/tex]R.
    [tex]\Rightarrow[/tex] Df=(2A)(0)-(0)2=0

    This is where I get stuck. Now that Df=0, how do I determine whether or not the critical pts are local extrema or saddle pts?

    From plotting the function on Mathematica, I know that these critical points are in fact saddle points, but I don't know how to mathematically state that.

    Thanks!
     
  2. jcsd
  3. Dec 10, 2009 #2

    Mark44

    Staff: Mentor

    Re: Finding the critical points of a multivariable function and determining local ext

    It seems to me that you are making this much harder than it needs to be by not sketching a graph of this surface. Since y doesn't appear explicitly in the formula for the function, this surface is a cylinder with parabolic cross section, and with its axis of symmetry in the direction of the y-axis. IOW, the surface looks something like a trough. If A > 0, the trough opens upward, and all critical points are global minima. If A < 0, the trough opens downward, and all critical points are global maxima. All critical points lie on a line that is parallel to the y-axis.
     
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