Answer
4096/25$\sqrt5$
Work Step by Step
f(x,y,z) = xyz
g(x,y,z) = x + y + z^{2} -16
∇f = yzi + xzj + xyk
∇g = i + j + 2zk
∇f = λ∇g
yzi + xzj + xyk = λ(i + j + 2zk)
yz = λ ,xz = λ , xy = 2zλ -- (1)
either z = 0 or y = x
As product can't be 0
z $\ne$ 0 , y = x
In (1)
x$^{2}$ = 2zλ , xz = λ
x$^{2}$ = 2z(xz)
x = 0 or x = 2z$^{2}$
x = 2z$^{2}$ , y = 2z$^{2}$ --(2)
2z$^{2}$ + 2z$^{2}$ + z$^{2}$ = 16
z = ± 4/$\sqrt 5$
Using this value in (2)
x = 32/5 and y = 32/5
f(32/5 , 32/5 , 4/$\sqrt5$) = 4096/25$\sqrt5$
