Question

2题第一问，谢谢

Answer 1

先求偏导:运用隐式方程求导法则.
z对x:4x+2z·偏z/偏x+8(z+x·偏z/偏x)-偏z/偏x=0
→偏z/偏x=-(4x+8z)/(2z+8x-1).
z对y:4y+2z·偏z/偏y2+8x·偏z/偏y-偏z/偏y=0.
→偏z/偏y=-4y/(2z+8x-1).
则方向导数z'=√[(偏z/偏x)^2+(偏z/偏y)^2]
=√[(4x+8z)^2+(4y)^2] /(2z+8x-1)
=4√[(x+2z)^2+y^2] /(2z+8x-1)
当方向导数z'=0时,函数z=z(x,y)取得极值.
则必有：(x+2z)^2+y^2=0
∴此时x+2z=0且y=0.
则x=-2z.
将x=-2z,y=0代入原隐式方程2x^2 +2y^2+z^2+8xz-z+8=0,得
8z^2 +0+z^2-16z^2-z+8=0
7z^2+z-8=0.
求得:z=1或-8/7.
即z=z(x,y)的
极大值是1;
极小值是-8/7.