Find the linearization of the function at the given point.
-f(x, y, z) = 8xy + 4yz + 8zx at (1, 1, 1)
A) L(x, y, z) = 8x + 4y + 8z - 20
B) L(x, y, z) = 16x + 12y + 12z - 20
C) L(x, y, z) = 16x + 12y + 12z - 40
D) L(x, y, z) = 8x + 4y + 8z - 40

Question

Quizplus · Accepted Answer

To find the linearization at (1, 1, 1), we need to first find the partial derivatives of f(x, y, z):
fx = 8y + 8z
fy = 8x + 4z
fz = 4y + 8x
Then, plug in the point (1, 1, 1) and simplify:
fx(1, 1, 1) = 8(1) + 8(1) = 16
fy(1, 1, 1) = 8(1) + 4(1) = 12
fz(1, 1, 1) = 4(1) + 8(1) = 12
So the linearization at (1, 1, 1) is:
L(x, y, z) = f(1, 1, 1) + fx(1, 1, 1)(x-1) + fy(1, 1, 1)(y-1) + fz(1, 1, 1)(z-1)
= (8(1)(1) + 4(1)(1) + 8(1)(1)) + 16(x-1) + 12(y-1) + 12(z-1)
= 16x + 12y + 12z - 20
Therefore, the best choice is B.

Find the Linearization of the Function at the Given Point