Question

Find the linearization L(x) of the function at x = a, f(x) = x(1/x), a = 1.

a) L(x) = 2 - x
b) L(x) = 1/x
c) L(x) = 0
d) L(x) = 1

Answer 1

The linearization of the function f(x) = x(1/x) at x=1 is L(x) = 1.

Step-by-step explanation:

The linearization of a function f(x) at x=a is given by:

L(x) = f(a) + f'(a)(x-a)

Here, the function f(x) is x(1/x) and a = 1. To find the linearization L(x) at x=1, we first need to find the value of f(1) and f'(1).

We know that f(x) = x(1/x) = 1, for x ≠ 0.

So, f(1) = 1(1/1) = 1. Now, let's find the derivative of f(x) to find f'(1).

f'(x) = d/dx(x(1/x)) = 1(1/x) + x(-1/x^2) = 1/x - 1/x^2

So, f'(1) = 1/1 - 1/1^2 = 1 - 1 = 0.

Substituting the values of f(1) and f'(1) in the linearization formula, we get:

L(x) = 1 + 0(x-1) = 1.

Therefore, the linearization of the function f(x) = x(1/x) at x=1 is L(x) = 1.