Science:Math Exam Resources/Courses/MATH103/April 2013/Question 05 (b)/Solution 1

Setting the derivative to 0 and solving gives

$0=4x^{3}-12x^{2}+8x=4x(x^{2}-3x+2)=4x(x-1)(x-2)$

and thus, the steady states are $x=0,1,2$. At the point a, the derivative is positive and so the function $x(t)$ is increasing and thus approaches the steady state 1 (the first steady states bigger than a). At b, the dervative is negative hence the function is negative and thus approaches the steady state 1. At c, the derivative is positive and so the function is increasing and as t approaches infinity, the function $x(t)$ approaches infinity.