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Math Help - Partial Derivatives

  1. #1
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    Partial Derivatives

    Clearly, this function has a minimum at (0,0) yet, it is not differentiable at (0,0). How then should I show that it has a minimum at (0,0)?

    z= \left( {x}^{2}+{y}^{2} \right) ^{ 0.5}

    First partial:

     \left( {x}^{2}+{y}^{2} \right) ^{- 0.5}x
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  2. #2
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    Quote Originally Posted by econstudent View Post
    Clearly, this function has a minimum at (0,0) yet, it is not differentiable at (0,0). How then should I show that it has a minimum at (0,0)?

    z= \left( {x}^{2}+{y}^{2} \right) ^{ 0.5}

    First partial:

     \left( {x}^{2}+{y}^{2} \right) ^{- 0.5}x
    Well, here's a simpler question to think about .... how would you show that y = |x| has a minimum at x = 0?
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    Quote Originally Posted by mr fantastic View Post
    Well, here's a simpler question to think about .... how would you show that y = |x| has a minimum at x = 0?
    y>0 for all non-zero x. Therefore, x = 0 is the minimum. You can have a piece-wise function that shows that y is always positive for x<0 and x>0.

    I guess you can use a limit?
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  4. #4
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    maxima minima for 2 variable functions

    Quote Originally Posted by econstudent View Post
    y>0 for all non-zero x. Therefore, x = 0 is the minimum. You can have a piece-wise function that shows that y is always positive for x<0 and x>0.

    I guess you can use a limit?
    1)find partial derivative of z w.r.t. x.
    2)equalise that to 0 and solve.
    3)similar process for z w.r.t y.
    4)you will get a point(s) (x,y) which will give either max or min values.
    5) fxx.fyy-(fxy)^2>0 for max or min points
    6)fxx>0 implies min value otherwise max value.

    fxx=partial derivative for 2 times each time wrt x
    fyy=partial derivative for 2 times wrt y
    fxy=partial derivative once wrt x then wrt y.
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