Question

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

Answer 1

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