1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]\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.

    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.

  2. jcsd
  3. Dec 10, 2009 #2


    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook