Answer
See proof below.
Work Step by Step
Let $s(t)$ be the function of the distance covered by time t, (s is in miles, t in hours.)
and let us assume that the function $s$ is continuous and differentiable over $[0,2].$
The function of speed is the derivative of s. Speed = $s'(t)$
We are given: $\left\{\begin{array}{l}
s(0)=0\\
s(2)=D
\end{array}\right., $ where D represents total distance.
By the Mean Value Theorem, there exists a moment in time $t_{0}\in(0,2)$
where $s'(t_{0})=\displaystyle \frac{s(2)-s(0)}{2-0}=\frac{D}{2}$ miles per hour
The expression $\displaystyle \frac{s(2)-s(0)}{2-0}=\frac{D}{2}$ is the average speed of the trip
(total distance) / (total time).
Thus, the car will have had the average speed at a certain moment $t_{0}\in(0,2)$.
