Applying the MVT to guarantee a specific instantaneous speed

The average velocity over the trip is $\frac{s(4)-s(0)}{4-0} = \frac{200-0}{4} = 50$ mph. Since the car's position function is continuous on $[0,4]$ and differentiable on $(0,4)$ (physically, velocity is always defined), the Mean Value Theorem applies on this interval.

The MVT states that there exists at least one $c$ in $(0,4)$ such that $s'(c) = \frac{s(4)-s(0)}{4-0} = 50$. Since $s'(t)$ represents the car's instantaneous velocity, this guarantees the car's speedometer must have read exactly 50 mph at some instant $c$ during the trip — the average velocity cannot be maintained without the instantaneous velocity equaling it somewhere. Geometrically, this means some tangent line to the position graph is parallel to the secant line joining $(0,0)$ and $(4,200)$.

The tangent line at the guaranteed instant is parallel to the secant line. This is because the Mean Value Theorem guarantees a point where the instantaneous rate of change equals the average rate of change over the whole interval. Since the secant line's slope represents the average velocity, a tangent line with the same slope must run parallel to it.