Question

Question #f6ebb

Answer 1

The linearization function is `L(x)=-9-16x`

For any function f(x), the linearization function L(x) at any point a can be found using the relation

`L(x)=f(a)+f'(a) (x-a)`
where f'(x) is the first derivative of the function f(x).

Basically, this formula for `L(x)` is the value of the function at the point a plus the slope of the function at the point a times any displacement from the point a.

Using information from the question:
`f(x)=x^4+6x^2` and `a=-1`
`f(a)=f(-1)=(-1)^4+6(-1)^2=1+6=7`
`f'(x)=4x^3+12x`
`f'(a)=f'(-1)=4(-1)^3+12(-1)=-4-12=-16`
`L(x)=f(a)+f'(a) (x-a)=7-16(x-(-1))`
which simplifies to
`L(x)=-9-16x`

As a check, we can verify that `L(a)=-9+16=7=f(a)`