Answer
max val = 1 + 6$\sqrt3$
min val = 1 - 6$\sqrt3$
Work Step by Step
f(x,y,z) = x$^{2}$yz + 1
g1(x,y,z) = z - 1
g2(x,y,z) = x$^{2}$ + y$^{2}$ + z$^{2}$ - 10
∇f = 2xyzi + x$^{2}$zj + x$^{2}$yz
∇g1 = k
∇g2 = 2xi + 2yi + 2zK
∇f = λ∇g1 + µ∇g2
2xyzi + x$^{2}$zj + x$^{2}$yk = λ(k) + µ(2xi + 2yj + 2zk)
2xyz = µ2x , x$^{2}$z = 2yµ , x$^{2}$y = λ + 2zµ
xyz = xµ
case 1 : x = 0 or z = 1
in g(1)
0$^{2}$ + y$^{2}$ + 1$^{2}$ - 10 = 0
y$^{2}$ = 9
y = ±3
(0,±3,1)
case 2 : y = µ and z = 1
x$^{2}$ = 2$y^{2}$
in g(1)
2y$^{2}$ + y$^{2}$ + 1$^{2}$ -10 = 0
3y$^{2}$ = 9
y = ±$\sqrt3$
x$^{2}$ = 2(±$\sqrt3$)$^{2}$
x = ±$\sqrt6$
(±$\sqrt6$,±$\sqrt3$,1)
f(0,±3,1) = 1
f(±$\sqrt6$,±$\sqrt3$,1) = 6(±$\sqrt3$) + 1 = 1 ± 6$\sqrt3$
max val = 1 + 6$\sqrt3$
min val = 1 - 6$\sqrt3$
