请找出函数f(x,y,z)=x^2*y^2*z^2受约束于x^2+y^2+z^2=1的最高(大)与最低(小)的点和值.原题为英文:findthepointsandvaluesoftheminimumandmaximumofthefunctionf（x,y,z）=x^2*y^2*z^2subjecttotheconstraintx^2+y^2+z^

Question

请找出函数f(x,y,z)=x^2*y^2*z^2 受约束于 x^2+y^2+z^2=1的最高(大)与最低(小)的点和值.
原题为英文:
find the points and values of the minimum and maximum of the function f（x,y,z）=x^2*y^2*z^2 subject to the constraint x^2+y^2+z^2=1

Answer 1

Lagrange乘子法：F=f(x,y,z)+a(x^2+y^2+z^2--1),
aF/ax=2xy^2z^2+2ax=0,
aF/ay=2yx^2z^2+2ay=0,
aF/az=2zx^2y^2+2az=0,三式分别乘以x,y,z相加得
6x^2y^2z^2+2a(x^2+y^2+z^2)=0,即f(x,y,z)=x^2y^2z^2=--a/3.因此求出a的最大值和最小值即可.
再由前面三个式子知道2ax^2=2ay^2=2az^2
当a不为0时,得x^2=y^2=z^2.故x^2=y^2=z^2=1/3,此时maxf(x,y,z)=1/27.
当a等于0时,x,y,z中必有一个是0,此时minf=0.
至于point,x^2=y^2=z^2对应有8个点,（正负根号(3)/3,正负根号(3)/3,正负根号(3)/3）.
x,y,z有一是0,无穷多个点.例如（0,y,正负根号(1-y^2)),自己去写其他的点吧.