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.