I am aware of what the discriminant ($b^2-4ac$) means in relation to a function $f(x)$ when referring to the number of real roots. $$\begin{align} b^2-4ac&=0& &\longrightarrow &&\text{ two equal roots} \\ b^2-4ac&>0& &\longrightarrow &&\text{ two distinct roots} \\ b^2-4ac&<0& &\longrightarrow &&\text{ no roots} \end{align}$$
However, does this change when referring to differential equations?
I read that $f'(x) > \text{or} = 0$ signifies that the $f'(x)$ has two equal roots or no roots (as written on the answer sheet, circled line)
Why is this the case?
(below are specific questions and their answers)
If $f'(x)\geq 0$ for all $x$, that means that either $f'(x)>0$ for all $x$ or $f'(x)\geq 0$ for all $x$. In the first case, the graph lies completely under/above the $x$-axis (therefore it has no roots). In the second case, the graph touches the $x$-axis (therefore has two equal roots).
If the translate this into discriminant-talk: we must have $$\Delta=b^2-4ac=4p^2-12pq\leq 0 \iff p^2\leq 3pq$$