※ 引述《j19880706 (smallpig)》之铭言： : Find the minimum of the function f(x,y,z)=xy+4yz+8xz : subject to the constraint xyz=256 : (The existence of the minimum is suggested by the geometry of the problem.) : 这题要怎麽算　要用微分吗 : 麻烦大大教教我 : 谢谢您 令 g(x,y,z) = xyz - 256 F(x,y,z) = f(x,y,z) + (λ)(g(x,y,z)) = xy + 4yz + 8xz + (λ)(xyz - 256) Fx = y + 8z + (λ)(yz) = 0 => -y - 8z = (λ)(yz) -y - 8z => λ = --------- ------(1) yz Fy = x + 4z + (λ)(xz) = 0 => -x - 4z = (λ)(xz) -x - 4z => λ = --------- ------(2) xz Fz = 4y + 8x + (λ)(xy) = 0 => -4y - 8x = (λ)(xy) -4y - 8x => λ = ---------- ------(3) xy (1) x y + 8z ----- => (---)(--------) = 1 (2) y x + 4z => (x)(y + 8z) = (y)(x + 4z) => xy + 8xz = xy + 4yz => 8xz = 4yz => y = 2x (z > 0) ------(4) (2) y x + 4z ----- => (---)(---------) = 1 (3) z 4y + 8x => (y)(x + 4z) = (z)(4y + 8x) => xy + 4yz = 4yz + 8xz => xy = 8xz => y = 8z (x > 0) ------(5) (4) x ----- => 1 = ---- => x = 4z (5) 4z x = 4z , y = 8z 代入 xyz = 256 得 (4z)(8z)(z) = 256 z^3 = 8 => z = 2 => x = 4z = 8 , y = 8z = 16 因此当 x = 8 , y = 16 , z = 2 时 , f(x,y,z) 有最小值 f(8,16,2) = (8)(16) + (4)(16)(2) + (8)(8)(2) = 128 + 128 + 128 = 384 --

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