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Math Help - Finding maxima and minima of two variable function

  1. #1
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    Finding maxima and minima of two variable function

    Find the local maximum and minimum values and saddle points of the function :

    f(x,y) = x^3 - 6xy + 8y^3

    I found the critical points (0,0) and (1,1/2) by setting the partial derivatives equal to zero and solving for x and y.

    Then, I tried the second derivative test:
    D = fxx(a, b) * fyy (a, b) - [fxy(a, b)]^2
    But found that D(0,0) = 0, meaning that the second derivative test fails for the point (0,0). What do I do when the second derivative test fails? Note that I am not allowed a graphing calculator.

    Thank you,
    David

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  2. #2
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    Quote Originally Posted by Dwill90 View Post
    Find the local maximum and minimum values and saddle points of the function :




    f(x,y) = x^3 - 6xy + 8y^3

    I found the critical points (0,0) and (1,1/2) by setting the partial derivatives equal to zero and solving for x and y.

    Then, I tried the second derivative test:
    D = fxx(a, b) * fyy (a, b) - [fxy(a, b)]^2
    But found that D(0,0) = 0, meaning that the second derivative test fails for the point (0,0). What do I do when the second derivative test fails? Note that I am not allowed a graphing calculator.

    Thank you,
    David

    f_x=3x^2-6y\,,\,f_{xx}=6x\,,\,\,f_y=-6x+24y^2\,,\,f_{yy}=48y,<br />
f_{xy}=f_{yx}=-6\Longrightarrow \left(f_{xx}f_{yy}(0,0)\right)-\left(f_{xy}(0,0)\right)^2=0-(-6)^2=-36 ...why do you say it is zero?

    Tonio
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  3. #3
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    I see, I miscalculated the partial derivative fxy, thinking it to be zero instead of -6. Thanks for the help!
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