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Math Help - Local Extrema help

  1. #1
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    Local Extrema help

    How do i find the "local extrema" for a function like this: f(x,y) = xy+y-15x
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  2. #2
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    Hello, square!

    How do i find the "local extrema" for a function like this: . f(x,y) \:=\: xy+y-15x
    Solve the system: . \begin{array}{ccc}\dfrac{\partial f}{\partial x} &=& 0 \\ \\[-3mm] \dfrac{\partial f}{\partial y} &=& 0 \end{array}

    And test the critical value(s) in: . D \;=\;\left(\frac{\partial^2\!f}{\partial x^2}\right)\left(\frac{\partial^2\!f}{\partial y^2}\right) - \left(\frac{\partial^2\!f}{\partial x\partial y}\right)^2


    If all this is meaningless to you,
    . . (1) you should not have been assigned this problem, or
    . . (2) you weren't paying attention in class.

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  3. #3
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    Quote Originally Posted by square View Post
    How do i find the "local extrema" for a function like this: f(x,y) = xy+y-15x

    Solve the equations f_x=y-15=0\,,\,\,f_y=x+1=0 , and then use the Hessian's determinant in the points (x,y) that you found above, f_{xx}f_{yy}-(f_{xy})^2 : if this determinant is positive and (1) f_{xx}>0 , then the point you found is a local minimum, and if (2) f_{xx}<0 the point is then a local maximum .

    If the determinant is negative then the point is a saddle point (not max. not min.), and if the determinant is zero then the Hessian matrix's test fails to decide.

    Tonio
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