Question

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

Answer 1

A linearization is a tangent line.

Explanation:

The linearization of `f(x) = x^4 + 4x^2` at `a=1` is one form of the equation of the line tangent to the graph of `f` at the point `(1,f(1))`.

For `f(x) = x^4 + 4x^2`, we have `f(1) = 5` and `f'(x) = 4x^3+8x`, so `f'(1) = 12`.

The tangent line has point-slope form:

`y-f(1) = f'(1)(x-1)` `" "` `" "` `y-5=12(x-1)`.

The linearization is:

`y=f(1) + f'(1)(x-1)` `" "` `" "` `y=5 + 12(x-1)`.