Answer:
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 +