※ 引述《iips (朝目标前进中)》之铭言： : Find the dimensions that will minimize the surface : area (and hence the cost) of a rectangular fish aquarium, : open on top,with a volume of 32 ft^3. : Ans:4 ft by 4ft for the base;2 ft for the height 令 f(x,y,z) = xy + 2xz + 2yz , g(x,y,z) = xyz - 32 F(x,y,z) = f(x,y,z) + (λ)(g(x,y,z)) = (xy + 2xz + 2yz) + (λ)(xyz - 32) Fx = y + 2z + (λ)(yz) = 0 => -y - 2z = (λ)(yz) -y - 2z => λ = --------- ------(1) yz Fy = x + 2z + (λ)(xz) = 0 => -x - 2z = (λ)(xz) -x - 2z => λ = --------- ------(2) xz Fz = 2x + 2y + (λ)(xy) = 0 => -2x - 2y = (λ)(xy) -2x - 2y => λ = ---------- ------(3) xy (1) x y + 2z ----- => (---)(--------) = 1 (2) y x + 2z => (x)(y + 2z) = (y)(x + 2z) => xy + 2xz = xy + 2yz => xz = yz => x = y (z > 0) ------(4) (2) y x + 2z ----- => (---)(---------) = 1 (3) z 2x + 2y => (y)(x + 2z) = (z)(2x + 2y) => xy + 2yz = 2xz + 2yz => xy = 2xz => y = 2z (x > 0) ------(5) 由(4)、(5)得 x = 2z , y = 2z 代入 xyz = 32 得 (2z)(2z)(z) = 32 => z^3 = 8 => z = 2 x = 2z = 4 , y = 2z = 4 因此当 x = 4 , y = 4 , z = 2 时 f(x,y,z)有最小值 f(4,4,2) = (4)(4) + (2)(4)(2) + (2)(4)(2) = 16 + 16 + 16 = 48 因此最小表面积为 48 --

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