FIRST HALF

**PSV Eindhoven** is **going home** from the **match** they played yesterday evening vs. **Feyenoord**.
After **sleeping** in **Rotterdam**, the bus has left **Rotterdam** at **10:00** a.m. road to **Eindhoven**.

At one point, **Mark Van Bommel**, who boasts about his **extraordinary memory**, says: **'Yesterday we went through this place just at this very hour'**.

His teammates do not quite believe him, but in his insistence, they decide to see if he's right. To confirm this, they remember that some **PSV** **supporters** where beside the **road** to wave to them at a place near the town of **Breda**, so they decide to call the president of the **PSV** **Supporters' Club of Breda**, in order to see if he knew **at what time** the bus had gone through **Breda** on Saturday and Sunday.

And indeed, the Club's president confirms that on both **Saturday** and **Sunday** they waved to the players in **Breda** at **10:57** hours.

**Van Bommel**'s teammates insist he's been very lucky, and propose him a **bet**, on whether he will be able to do the same in **all journeys** of the Dutch league (**Eredivisie**).

Knowing that **Van Bommel** han an excellent **memory**, able to recall places and related times without problem, and that **PSV** **bus** always begins its **outward journeys** at **10:00** a.m., they sleep in the match town, and also start the **return journey** at **10:00** a.m. on the **next morning**, do you think that **Van Bommel** should accept the **bet**?

In other words, do you think that **in every journey** there will be a **place** for which the bus goes through at the **same time** on both the **outward** and the **return** journeys?

A priori, it doesn't seem logical that the bus necessarily have to go through **the same point** at the **same time** on the outward and the return journeys, even though both departs are at the same hour.

In fact, there are **numerous factors** that influence every journey: the bus can go **faster**, or **slower**, or stop to filling up with **fuel**. Sometimes it will stop for a **break**, or because of a tyre puncture, sometimes it will find a **traffic-jam** by an accident ...

Therefore, it seems quite improbable that there should be a point in the road where the bus goes through both days at exactly the same time.

And nevertheless, it **will** happen **every** journey, as you'll see. If we are to address this **problem** in a direct way, we'll find many difficulties. Therefore, in this case we should study the problem from **another angle**, from which the problem seems really simple.

We can think of **two buses** leaving at the **same time**, one from **Eindhoven** and the other one from **Rotterdam**. This view does not affect the conditions of the problem, nor at what time the bus will pass through each point of the route. Thus, we can see clearly that there will be a moment when **the two buses necessarily will meet** somewhere in the road, that is, the two buses will be at the same time in the same place.

And this will happen **every time**. Sometimes the place will be closer to home, and sometimes closer to the destination, sometimes it will be sooner, and others will be later. But there will be always a cross in their way.

That's what happens with **PSV** **bus**, which is **only one bus**, but traveling on **different days**. On every journey back there will be a place at which the bus was going through the previous day at the same time.

So **Mark Van Bommel** should to take advantage of his extraordinary memory and **accept** his teammates **bet**.

A more **academic** **analysis** of this problem leads us to solve it by using the **Intermediate value theorem**. This theorem states that if a function * f* is continuous on a closed nonempty interval

We can **solve it** in an even more elegant way with **Bolzano**'s **theorem **, which is a particular case of the Intermediate value theorem. This theorem states that if * f* is a continuous function on the interval

In our particular case, we define ** f(x)** as the function that indicates the km. number of the road in which the bus is at time

We also define ** g(x)** as the function that indicates the km. number of the road in which the bus is at time

And finally we define function ** h(x) = f(x) - g(x)** on the interval

That way we **demonstrate** that in outward and return journeys, there's **always** a kilometer point ** f(c)** where the bus goes through at the same time

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