## Calculus (3rd Edition)

(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.