Mean/intermediate value theorem

Protonics · New Member Offline Joined: 20 Sep 2009 Posts: 8

A and B are two points on the motor way, and they lie 75 km apart. Two cars pass point A at 10 pm and point B at 11 pm. Show that the cars must have had exactly the same speed at atleast one point in the time interval.
And: How much information do you actually need to be able to solve this task?

Thanks!

**Posted:** Mon Nov 02, 2009 5:33 am

AndrewTom

Hi. I'd use Rolle's theorem (see here: http://en.wikipedia.org/wiki/Rolle's_theorem ) .

Let $f(t)$ and $g(t)$ be the distances travelled by the cars in time $t$ , $\ 0 \le t \le 1$ , so that their speed are $f'(t)$ and $g'(t)$ .

$f(0) = g(0) = 0$ and $f(1) = g(1) = 75$ .

If we let $h(t) = f(t) - g(t)$ then $h(0) = h(1) = 0$

and so by Rolle's theorem, there exists $a$ , with $0 < a < 1$ , such that $h'(a) = 0$ ,

that is $f'(a) - g'(a) = 0$ ,

that is $f'(a) = g'(a)$ .

The actual times they pass $A$ and $B$ , provided they're the same times, are not needed; nor is the actual distance between $A$ and $B$ .

**Posted:** Mon Nov 02, 2009 10:06 am