Mean Value Theorem Practice Test

No, because knowing endpoint values alone never guarantees any derivative value.

Yes, because $f$ is continuous on $[1,5]$ and differentiable on $(1,5)$, so some $c$ has $f'(c)=\dfrac{10-2}{5-1}=2$.

Yes, because $f(5)-f(1)=8$, so there must be a point where $f'(c)=8$.

Yes, because $f$ is continuous on $[1,5]$, so $f'(c)$ must equal the average rate of change.

Yes, because $f(1)$ and $f(5)$ are positive, so $f'(c)=2$ for some $c$.