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Math Help - second derivative along a path

  1. #1
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    second derivative along a path

    Hello,
    My question is :

    Show that the function
    f(x,y) = xy has a saddle point at (x,y) = (0,0) and find the second derivative along a path through that point has the greatest positive and negative values.

    My thoughts:

    If I substitute y = x into the function as thus f(y,y) = x.x = x^2 and then find the second derivative as thus d^2f/dx^2 = 2 this is repeated for the path y = -x as thus f(y,y) = -x.x = -x^2with a second derivative of d^2f/dx^2 = -2

    This is repeated for paths x=y and x=-y giving a derivative of d^2f/dy^2 = 2 and d^2f/dy^2 = -2 repectively. So the greatest positive and negative values are 2 and -2. At y=0 and x = 0 the function is constant.

    Any Help would be greatly appreciated.
    Thank you,
    Riptorn70.
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  2. #2
    MHF Contributor

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    I don't understand your reasoning. You calculated the second derivatives along two different paths and got -2 and 2. What makes you sure that those are the smallest and largest values?

    What if we approach (0, 0) along the line y= 10000x? Then f(x,y)= f(x,10000x)= 10000x^2. What is the second derivative of that?

    Do it more generally. Since the derivative is "local", we can approximate any path through (0, 0) by a straight line: y= mx for some number, m. Then f(x,y)= f(x, mx)= mx^2. What is the second derivative of that?

    That includes every path except vertical ones- on a line tangent to such a path, at (0, 0), x= 0 so f(x,y)= f(0, y)= 0 and the second derivative is 0.
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  3. #3
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    thank you hallofIvy!!
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