Let $y(x)$ is the solution of the D.E. $y' = y(y-2); y(0)=\alpha,$ then-
(a) $y(x)$ is increasing for all $\alpha$
(b) $y(x)$ is decreasing for all $\alpha$.
(c) $y(x)$ is increasing if $\alpha \in (-\infty, 0) \cup (2, \infty)$
(d) $y(x)$ is decreasing if $\alpha \in (0, 2)$
Explanation: (c) and (d) are correct.
Critical points of $y(x)$ are given by $y'=0$ i.e $y(y-2)=0$. Thus $y=0, y=2$ are critical points.
$\implies \dfrac{dy}{dx} > 0$ if $(-\infty, 0) \cup (2, \infty)$, and $\dfrac{dy}{dx} < 0$ if $y \in (0, 2)$
$\implies$ (c) and (d) are correct.