On a recent problem, I received the following scenario: An object moves back and forth on a straight track. During the time interval $0\le t\le 30$ minutes, the object's position, $x$, and velocity, $v$, are continuous functions; some of their values are shown in the table (which I have reproduced below).

$\begin{array}{|ccc|}\hline t\mathbf{\text{(min)}}& x(t)\mathbf{\text{(feet)}}& v(t)\mathbf{\text{(feet/min)}}\\ 0& \text{12}& -20\\ 10& \text{50}& \text{20}\\ 15& \text{18}& \text{3}\\ 25& \text{60}& -2\\ 30& \text{60}& \text{10}\\ \hline\end{array}$

The question was, for $0<t<30$, does there exist a time t when $v(t)=-22$?

I tried to apply the intermediate value theorem, and concluded that the answer was "not necessarily." I reasoned that because the values of $v(t)$ in the table ranged between −20 (at the beginning) and 10 (at the end), and −22 wasn't between those two values, we couldn't be sure that such a t exists.

The teacher gave me 3/4 points on the question. Their comment was that I should have considered the mean value theorem, too. They did not write anything about my intermediate value theorem analysis, but I still have grave doubts about my application of the intermediate value theorem ... and, as for the mean value theorem, I have no idea how to proceed.

Would anyone here be able to shed some light on how the intermediate value and mean value theorems could be used to determine whether there exists a t where $v(t)=-22$? Thanks so much.