高数题目求解答
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构造拉格朗日函数 F = 2xy+2xz+yz+λ(xyz-k)
Fx = 0 : 2y+2z+λyz = 0 (1)
Fy = 0 : 2x+z+λxz = 0 (2)
Fz = 0 : 2x+y+λxy = 0 (3)
Fλ = 0 : xyz = k (4)
y(2) - z(3) : 2x(y-z) = 0, 得 z = y
x(1) - y(2) : z(2x-y) = 0, 得 y = 2x,
将 z = y = 2x 代入 (4) ，2x^3 = k,
得 x = (k/2)^(1/3), y = z = 2(k/2)^(1/3)
表面积最小值 S = 2xy+2xz+yz = 8x^2 = 8(k/2)^(1/3) = 4(2k^2)^(1/3)
Fx = 0 : 2y+2z+λyz = 0 (1)
Fy = 0 : 2x+z+λxz = 0 (2)
Fz = 0 : 2x+y+λxy = 0 (3)
Fλ = 0 : xyz = k (4)
y(2) - z(3) : 2x(y-z) = 0, 得 z = y
x(1) - y(2) : z(2x-y) = 0, 得 y = 2x,
将 z = y = 2x 代入 (4) ，2x^3 = k,
得 x = (k/2)^(1/3), y = z = 2(k/2)^(1/3)
表面积最小值 S = 2xy+2xz+yz = 8x^2 = 8(k/2)^(1/3) = 4(2k^2)^(1/3)
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