How do you find the linearization of the function at a specific point, say x = 12, for a function given as f(x) = 9?

Finding the Linearization of a Function

To find the linearization of a function f(x) at a specific point a, we use the formula:

L(x) = f(a) + f'(a)(x – a)

Step-by-step Process:

1. Identify the function:

   In this case, we have f(x) = 9. This is a constant function.

2. Determine the point of linearization:

   We want to linearize at x = 12.

3. Calculate f(a):

   Since f(x) = 9, we have f(12) = 9.

4. Find the derivative f'(x):

   For our function, the derivative is f'(x) = 0 because the function is constant.

5. Evaluate f'(a):

   Thus, f'(12) = 0.

6. Substituting into the linearization formula:

   Now, we can substitute these values into our linearization formula:

   L(x) = f(12) + f'(12)(x – 12)

   This gives us:

   L(x) = 9 + 0(x – 12)

   which simplifies to:

   L(x) = 9

Conclusion:

The linearization of the function f(x) = 9 at the point x = 12 is simply L(x) = 9, which is consistent since the original function is constant.