# Mean Value Theorem proof question

What's the formal way to prove this? I drew a displacement/time graph with 2 random squiggles from the origin to the same point $(t,f(t))$, assuming that f(t) is where the finish line is.

Running through the MVT, for the hare:

$\dfrac{f(t)-0}{t-0}=f'(c)$

Since the end point and start point for the tortoise is the same, the gradient must be the same at some point.

As an instructor I wouldn't say your answer is really correct. Call $s_1$ and $s_2$ the two laws of motion, and notice that $s=s_1-s_2$ satisfies the assumption of Rolle's theorem. Then, at some time $t$, $s'(t)=0$, and you conclude that $s_1'(t)=s_2'(t)$.