用拉格朗日乘数法求内接于半径为A的球面且有最大体积的长方体…跪球答案啊 谢谢

设长方体的3个棱长分别为2x,2y,2z,
则x,y,z>0, x^2+y^2+z^2=A^2.
长方体的体积f(x,y,z)=(2x)(2y)(2z)=8xyz.

设F(x,y,z)=8xyz+a(x^2+y^2+z^2-A^2),
令
DF/Dx【F对x的偏导数】 = 8yz + 2ax = 0,
DF/Dy = 8xz + 2ay = 0,
DF/Dz = 8xy + 2az = 0,
x^2 + y^2 + z^2 - A^2 = 0.

8yz*8xz*8xy=-(2ax)(2ay)(2az),
64(xyz)=-a^3,
xyz=-a^3/64,a<0.
-2ax = 8yz = 8xyz/x = 8(-a^3/64)/x = -a^3/(8x),x^2 = a^2/16,x=-a/4,
-2ay = 8xz = 8xyz/y = 8(-a^3/64)/y = -a^3/(8y),y^2 = a^2/16,y=-a/4,
-2az = 8xy = 8xyz/z = 8(-a^3/64)/z = -a^3/(8z),z^2 = a^2/16,z=-a/4,
A^2 = x^2 + y^2 + z^2 = 3(-a/4)^2, a = -4A/3^(1/2),
x=y=z=-a/4=A/3^(1/2).

当长方体是边长为2A/3^(1/2)的正方体时，体积最大。