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

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

1. Identify the function and point: First, confirm the function you want to linearize and the point a at which you want to perform the linearization.
2. Calculate the derivative: Obtain the first derivative of the function, f'(x). This will allow you to determine the slope of the tangent line at point a. The derivative is given as follows:

f'(x) = 4x^3 + 4x

3. Evaluate the function and derivative at point ‘a’: Next, substitute a into both the function and its derivative to find f(a) and f'(a).
4. Formulate the linearization: The linearization L_x at point a is expressed using the formula:

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

5. Substitute values: Finally, replace f(a) and f'(a) in the linearization formula to write the linearization function.

This entire process results in a linear approximation of the function around the point a, which can be useful for various applications in calculus and mathematical analysis.