Best response correspondences game theory I'm studying game theory and there is a slide where I don't quite understand how exactly each value was deduced particularly:

For $BR_L(q)$ and $BR_c(q)$. I understand why they are functions of $q$ and $p$, since the best response of one player is dependent on what strategy the other player will choose, but i'm not sure exactly why they take values of $1$ and $[0,1]$ and $0$ for the specific values of $q$ and $p$. If someone could explain the derivation of these, that would help a lot!
Let $\pi_L(p,q) = 3pq + p(1-q) + 2(1-p)(1-q)$ denote the expected payoff of the lecturer. The best response is defined as $BR_L(q) = \arg\max_{p \in [0,1]}\pi_L(p,q)$.
Now, note that 
\begin{align}
\frac{\partial \pi_L(p,q)}{\partial p}
\begin{cases}
&gt; 0 \quad&amp; \text{for } q &gt; 1/4,\\
= 0 &amp; \text{for } q = 1/4,\\
&lt; 0 &amp; \text{for } q &lt; 1/4.
\end{cases}
\end{align}
For $q&gt;1/4$ the payoffs are strictly increasing in $p$ such that $p = 1$ is optimal.
For $q=1/4$ the payoffs are constant in $p$ such that any $p \in [0,1]$ is optimal.
For $q&lt;1/4$ the payoffs are strictly decreasing in $p$ such that $p = 0$ is optimal.
Same logic applies to The Catalan.