Step 1: Plan:
Think of the largest possible growth: if the slope never exceeds $5$, then in a distance of $4$ units the function cannot climb more than $20$.
Step 2: Steps:
Start at $f(1) = 4$. Over the interval $[1,5]$ of length $4$, the total rise is at most $5\times4 = 20$. So $f(5)\le4+20 = 24$.
Options such as $25$ are not possible, since they would need an average slope of $5.25$.
Final Answer:
The maximum value of $f(5)$ is $24$, option (C).
\[ \boxed{24} \]