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Math Help - relative extrema

  1. #1
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    relative extrema

    Hello,
    Locate all relative extrema and saddle points f(x,y)=6xy - x^6 - y^6.
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  2. #2
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    Hello, m777!

    I assume you know the procedure.
    . . Exactly where is your difficulty?


    Locate all relative extrema and saddle points: .f(x,y) .= .6xy - x^6 - y^6

    Solve the system: .f
    x = 0, fy = 0

    . . f
    x .= .6y - 6x^5 .= .0 . . y .= .x^5 . [1]
    . . f
    y .= .6x - 6y^5 .= .0 . . x .= .y^5 . [2]

    Substitute [1] into [2]: .x .= .(x^
    5)^5 . . x .= .x^25

    . . x^
    25 - x .= .0 . . x(x^24 - 1) .= .0 . . x .= .0, 1

    Then: .y .= .0, 1

    . . The critical points are: .(0,0,0), (1,1,4), (-1,-1,4)



    Second Partials Test: . f
    xx = -30x^4 . . fyy = -30y^4 . . fxy = 6

    . . Then: .D .= .(-30x^
    4)(-30y^4) - 6 .= .900x^4y^4 - 36


    At (0,0,0): .D .= .90000 - 36 .= .-6 . . . negative: saddle point at (0,0,0)


    At (1,1,4): .D .= .90011 - 36 .= .+864 . . . positive: extreme point
    . . f
    xx .= .-301^4 .= .-30 . . . negative: maximum at (1,1,4)


    At (-1,-1,4): .D .= .900(-1)^
    4(-1)^4 - 36 .= .+864 . . . positive: extreme point
    . . f
    xx .= .-30(-1)^4 .= .-30 . . . negative: maximum at (-1,-1,4)

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