How do you find the linearization L(x) of the function f(x) = x^4 – 2x^2 + 1?

To find the linearization L(x) of the function f(x) = x^4 – 2x^2 + 1 at a point a, we will use the formula for linearization:

L(x) = f(a) + f'(a)(x – a)

First, we need to determine the value of f(a):

f(a) = a^4 – 2a^2 + 1

Next, we need to find the derivative of f(x):

f'(x) = 4x^3 – 4x

Now, evaluate the derivative at the point a:

f'(a) = 4a^3 – 4a

Now we can plug these values back into the linearization formula:

L(x) = (a^4 – 2a^2 + 1) + (4a^3 – 4a)(x – a)

So, the linearization of f(x) at the point a is:

L(x) = a^4 – 2a^2 + 1 + (4a^3 – 4a)(x – a)