Answer
(a) sometimes true.
(b) true.
(c) never true.
(d) sometimes true.
Work Step by Step
(a) sometimes true. Since $ f(c)=3$ and $3\notin (20,100)$, then according to IVT there is no $ c\in [1,5]$ such that $ f(c)=3$. However, $f(c)=3$ may still exist (the IVT simply does not guarantee it).
(b) always true. Since $ f(c)=75$ and $75\in (20,100)$, then according to IVT there is $ c\in [1,5]$ such that $ f(c)=75$.
(c) never true. Since $ f(c)=50$ and $50\in (20,100)$, then according to IVT there is $ c\in [1,5]$ such that $ f(c)=50$.
(d) sometimes true. Since $ f(c)=30$ and $30\in (20,100)$, then according to IVT there is $ c\in [1,5]$ such that $ f(c)=30$ and $ c $ may be unique.
