How do you find the linearization of the function f(x) = x^4 – 6x^2 + 1 at a specific point?

Finding the Linearization of the Function

To find the linearization of the function f(x) = x^4 – 6x^2 + 1 at a specific point a, we can follow a systematic approach involving calculus, specifically the concept of derivatives.

Step 1: Choose the Point

First, we need to determine the point a at which we want to linearize the function. For the sake of our explanation, let’s say we choose a = 0.

Step 2: Find f(a)

Next, we calculate the value of the function at that point:

f(0) = (0)^4 - 6(0)^2 + 1 = 1

Step 3: Calculate the Derivative

Now we need to find the derivative of the function f(x) which is:

f'(x) = 4x^3 - 12x

To find the slope of the tangent line at the point a = 0, we evaluate the derivative at this point:

f'(0) = 4(0)^3 - 12(0) = 0

Step 4: Write the Linearization Formula

The formula for the linearization L(x) of a function around the point a is given by:

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

Step 5: Plug in Values

Now substituting our calculated values:

L(x) = f(0) + f'(0)(x - 0) = 1 + 0(x - 0) = 1

Conclusion

Therefore, the linearization of the function f(x) = x^4 – 6x^2 + 1 at the point a = 0 is simply:

L(x) = 1

This linearization indicates that at the point a = 0, the function behaves like a horizontal line.