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    • Thread Starter

    Right. I have notes that tell me one thing, and an example which contradicts it. Again, probably my sloppy note taking in the lecture, or maybe the notes are wrong... wondering if you can help me find out which!

    I'll run you through the general situation first:

    So we have a function  f(x,y) and we want to find it's maxima and minima. So we differentiate w.r.t x, then w.r.t y, and set the results to 0. This gives us a stationary point (a,b). Now we have to find the nature of it. So we take a hessian matrix and find it's determinant:

     detH = \left| \begin{array}{ccc}

f_{xx} & f_{xy} \\

f_{yx} & f_{yy} \end{array} \right| = 0

    The Hessian determinant is equal to 0, so we have no information and need further investigation!

    So we take function  \Delta(h,k) = f(a+h,b+k) - f(a,b) and use this result to find the nature of the stationary point.

    This is all fine and well. But in my notes it says this:


     \Delta(h,k) > 0 Then you have a local minimum at (a,b).

     \Delta(h,k) < 0 Then you have a local maximum (a,b)."

    However, during an example question we got the result  \Delta(h,k) > 0, and the conclusion was that we had a GLOBAL minimum at (a,b), not a local...

    So which is true? Does the  \Delta(h,k) give information about GLOBAL or LOCAL maxima/minima?


    local, but various other considerations could lead you to think a local minimum is a global minimum (which may be what was implicit in the lecture)
    • Thread Starter

    Ahhh... I've just realised that in this particular problem there was only one stationary point...

    Hence Global.

    Right, thankyou.


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