Question

consider the two equations below, each having the critical points a, b, and c, suppose that a < b < c. for one of these equations, only the critical point b is stable, for the other equation, b is the only unstable critical point. construct phase diagrams for the two equations to determine which is which. without attempting to solve either equation explicitly, make rough sketches of typical solution curves for each. you should see two funnels and a spout in one case, two spouts and a funnel in the other.
\\(\frac{dx}{dt}=(x - a)(x - b)(x - c)\\) \\(\frac{dx}{dt}=(a - x)(b - x)(c - x)\\
construct the phase diagram for \\(\frac{dx}{dt}=(a - x)(b - x)(c - x)\\). choose the correct answer below.
a.
stable unstable stable
\\(x<0\\ x = a\\ x>0\\ x = b\\ x<0\\ x = c\\ x>0\\
b.
stable unstable stable
\\(x>0\\ x = a\\ x<0\\ x = b\\ x>0\\ x = c\\ x<0\\
c.
unstable stable unstable
\\(x<0\\ x = a\\ x>0\\ x = b\\ x<0\\ x = c\\ x>0\\
d.
unstable stable unstable
\\(x>0\\ x = a\\ x<0\\ x = b\\ x>0\\ x = c\\ x<0\\)

Explanation:

Step1: Analyze sign of $\frac{dx}{dt}$

Let $y=(a - x)(b - x)(c - x)$. Critical points are $x = a,x = b,x = c$ with $a\lt b\lt c$. Consider intervals $(-\infty,a),(a,b),(b,c),(c,\infty)$.
For $x\lt a$, all of $(a - x),(b - x),(c - x)$ are positive, so $\frac{dx}{dt}=(a - x)(b - x)(c - x)>0$.
For $a\lt x\lt b$, $(a - x)<0,(b - x)>0,(c - x)>0$, so $\frac{dx}{dt}<0$.
For $b\lt x\lt c$, $(a - x)<0,(b - x)<0,(c - x)>0$, so $\frac{dx}{dt}>0$.
For $x>c$, $(a - x)<0,(b - x)<0,(c - x)<0$, so $\frac{dx}{dt}<0$.

Step2: Determine stability

A critical - point is stable if $\frac{dx}{dt}$ changes sign from positive to negative as $x$ increases through the point, and unstable if $\frac{dx}{dt}$ changes sign from negative to positive as $x$ increases through the point.
At $x = a$, $\frac{dx}{dt}$ changes from positive to negative, so $x = a$ is stable.
At $x = b$, $\frac{dx}{dt}$ changes from negative to positive, so $x = b$ is unstable.
At $x = c$, $\frac{dx}{dt}$ changes from positive to negative, so $x = c$ is stable.

Answer:

A. Stable, Unstable, Stable at $x = a,x = b,x = c$ respectively with $x'<0$ for $x\in(-\infty,a)\cup(c,\infty)$ and $x'>0$ for $x\in(a,b)\cup(b,c)$