Linearization of an equation around fixed points

[darkspym7]

Homework Statement

Find the linearization of the equation y' = y(-1+4y-3y^2) about each of the fixed points

The Attempt at a Solution

I think this is correct for finding fixed points:
Set y' = 0 = y(-1+4y+3y^2), so the fixed points are y = 0, 1/3, 1

What exactly does it mean by linearization of the equation around each of the fixed points?

 

[stakehoagy]

linearization

the linearization of a function f about a, (linearization at x=a) is
L(x)= f(a)+f'(a)(x-a) Its pretty much like a taylor series approximation.

 

[stakehoagy]

example

find the linearization of f(x)=x^2 about x=3

L(x)=f(3)+f'(3)(x-3)
L(x)=9+6(x-3)=6x-9

 

[darkspym7]

But using those fixed points, f'(a) would always be 0. Are those the correct fixed points?