Answer
4/3
Work Step by Step
f(x,y,z) = x$^{2}$ + 2y - z$^{2}$
g1(x,y,z) = 2x - y
g2(x,y,z) = y + z
∇f = 2xi + 2j
∇g1 = 2i - j
∇g2 = j + k
∇f = λ∇g1 + µ∇g2
2xi + 2j - 2zk = λ(2i-j) + µ(j+k)
2x = λ 2 , 2 = -λ + µ , -2z = µ
2x = λ2 , 2 = -λ + µ , -2z = µ
x = λ , 2 = -λ + µ , µ = -2z
2 = -x -2z
x = -2(1+z) -- substituting in g1
2(-2(1+z)) - y = 0 , y + z = 0
-4 + (-4z) - y = 0 , y + z = 0
4z + y = -4 , y = -z
4z - z = -4
3z = -4
z = -4/3
y = 4/3
x = -2(1+(-4/3))
x = 2/3
f(2/3,4/3,-4/3) = (2/3)$^{2}$ + 2(4/3) - (-4/3)$^{2}$
= 12/9
= 4/3 = max value
