How do you find the linearization of the function f(x) = x^4 + 2x^2 + x + 1 at a specific point?

To find the linearization L(x) of the function f(x) = x^4 + 2x^2 + x + 1 at a specific point, we will follow these steps:

1. Choose the point of linearization: Let’s assume we want to find the linearization at the point a. For example, let’s take a = 0.
2. Compute the function value: Calculate f(a):
   • f(0) = 0^4 + 2(0^2) + 0 + 1 = 1.
3. Find the derivative: Next, we need to compute the derivative f'(x):
   • Using the power rule, we find:
     • f'(x) = 4x^3 + 4x + 1.
4. Compute the derivative at the point: Now, substitute a into the derivative:
   
   f'(0) = 4(0^3) + 4(0) + 1 = 1.
5. Form the linearization: The linearization of the function at the point a is given by the formula:
   L(x) = f(a) + f'(a)(x – a).
   Substituting our values gives:
   L(x) = 1 + 1(x – 0) = x + 1.