limit22 發問時間: 科學數學 · 1 0 年前

微積分極大極小值題目

Find the local maximum and minimum values and saddle points

of f(x.y)=x^2 +y^2 +1/((x^2)(y^2)) x.y不等於0

1 個解答

評分
  • 天助
    Lv 7
    1 0 年前
    最佳解答

    Grad(f)=(∂f/∂x, ∂f/∂y)=(2x-2/(x^3 y^2), 2y-2/(x^2 y^3))=(0,0)

    then x^2=y^2=1,

    so, (x,y)=(1,1),(1,-1),(-1,1),(-1,-1)

    matrix of the second derivatives D=

    [ 2+ 6/(x^4 y^2) 4/(x^3y^3) ]

    [ 4/(x^3 y^3) 2+6/(x^2 y^4) ]

    at the critical points, D=

    [ 8 4 ] or [ 8 -4 ]

    [ 4 8 ] [-4 8 ]

    both are positive definite ( 8>0, det(D)>0)

    so f(1, 1)=f(1,-1)=...=f(-1,-1)= 3 are minimum values (no max. ...)

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