Question

8
Find the linearization of \( f(x, y, z)=x^{2}-x y+3 z \) at the point \( (2,1,0) \).

Answer 1

The linearization of a multivariable function f(x, y, z) at a point (a, b, c) is given by:

L(x, y, z) = f(a, b, c) + ∇f(a, b, c) · (x - a, y - b, z - c)

where ∇f(a, b, c) is the gradient of f at (a, b, c).

In this problem, we have:

f(x, y, z) = x² - xy + 3z

(a, b, c) = (2, 1, 0)

First, we find the gradient of f at (a, b, c):

∇f(a, b, c) = ⟨∂f/∂x, ∂f/∂y, ∂f/∂z⟩ evaluated at (a, b, c)

= ⟨2a - b, -a, 3⟩ evaluated at (2, 1, 0)

= ⟨3, -2, 3⟩

Next, we plug in the values for f(a, b, c), ∇f(a, b, c), x-a, y-b, and z-c:

L(x, y, z) = f(a, b, c) + ∇f(a, b, c) · (x - a, y - b, z - c)

= (2² - 2 +