Answer:
a) ∇f(x, y, z) = (2xyz − yz⁸, x²z − xz⁸, x²y − 8xyz⁷)
b) ∇f(6, −1, 1) = (-11, 30, 12)
c) Duf(6, −1, 1) = 84/5
Step-by-step explanation:
Given
f(x, y, z) = x²yz − xyz⁸
P(6, −1, 1)
u = (0, 4/5, − 3/5)
a) ∇f(x, y, z) = ?
We apply
fx(x, y, z) = 2xyz − yz⁸
fy(x, y, z) = x²z − xz⁸
fz(x, y, z) = x²y − 8xyz⁷
then
∇f(x, y, z) = (2xyz − yz⁸, x²z − xz⁸, x²y − 8xyz⁷)
b) ∇f(6, −1, 1) = (2*6*(-1)*1 − (-1)(1)⁸, (6)²(1) − (6)(1)⁸, (6)²(-1) − 8(6)(-1)(1)⁷)
⇒ ∇f(6, −1, 1) = (-11, 30, 12)
c) Duf(6, −1, 1) = ∇f(6, −1, 1)*(0, 4/5, − 3/5) = (-11, 30, 12)*(0, 4/5, − 3/5)
⇒ Duf(6, −1, 1) = 0 + 24 - 36/5 = 84/5
