太空總署 asked in 科學數學 · 1 decade ago

一題有關微積分解答的問題

請幫我解答~非常感激

Find the maximum, minimum, amd saddle points of f ( x, y ), if any, given that f ( x, y ) = x^3 - y^3 - 2xy + 6

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  • tom
    Lv 6
    1 decade ago
    Favorite Answer

    利用二階偏導數檢驗其極值及鞍點(saddle point)

    f(x,y) =x3 - y3 - 2xy+ 6

    fx = 3x2 - 2y ,fxx = 6x ,fxy = - 2

    fy = - 3y2 - 2x ,fyy = - 6y ,fyx = - 2

    設{fx = 3x2 - 2y= 0

    …{fy = - 3y2 - 2x= 0

    解上式聯立方程式得 (x,y) = (0,0)、(- 2/3,2/3) ……(臨界點)

    對於點 (0,0)

    △=|fxx(0,0) …fxy(0,0) |…|0… - 2|

    ……|fyx(0,0) …fyy(0,0) |=|- 2… 0|= - 4 < 0

    所以 (0,0) 為f(x,y) 的鞍點,f(x,y) 在 (0,0) 無相對極值

    對於點 (- 2/3,2/3)

    △=|fxx (- 2/3, 2/3) …fxy (- 2/3, 2/3)|…|- 4… - 2|

    ……|fyx (- 2/3, 2/3) …fyy (- 2/3, 2/3)|=|- 2… - 4|= 12 > 0

    而fxx (- 2/3, 2/3) = - 4 < 0

    所以f(x,y) 在 (- 2/3,2/3) 有相對極大值f(- 2/3,2/3) = 6又8/ 27

    《 有問題再提出 我會在意見欄回答》

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