a cyclist travels 30 km in one hour a cyclist travels 30 km in one hour
1) Demonstrate that there is an interval of 10 minutes such that the cyclist has traveled 5 Km
2) Is there always a time interval of 40 minutes during which he will have traveled 20 Km?
for the first question, that's what I did :
Note f the function that represents the distance traveled by the cyclist as a function of time (expressed in minutes). $f$ is a continuous function, $f (0) = 0$ and $f (60) = 30$. We want to prove that there is a time interval of 10 minutes such that the cyclist has traveled 5 km. In other words, we want to find $x \in [0,50]$ such that $f (x + 10) -f (x) = 5$.
Suppose that such an  $x$ does not exist. By the intermediate value theorem, we know that:
or else, 
$\forall x∈ [0,50]$, we have $f(x + 10) -f (x)> 5$.
or, $\forall x∈ [0,50]$, we have $f (x + 10) -f (x) <5$.
In the first case, we have
$f (60) =  f (60) -f (50) + f (50) -f (40) + f (40) -f (30) + f (30) -f (20) + f (20 ) -f (10) + f (10)-f(0) > 6 × 5=30$.
which is a contradiction. In the second case, we would find $f (60) <30$, which is also a contradiction. The hypothesis formulated is therefore false: there exists $x∈ [0,50]$ such that $f (x + 10) -f (x) = 5$.
For the second question,I Think that the answer in No but i don't know how to prove it .
Do you have any suggestions ? 
 A: I think that I found an answer for the second : if we prove that there is a time interval of 40 minutes where he would travel less than 20 km , it's done . 
Suppose that the cyclist traveled 15 km in the first 10 minutes then stopped for 40 minutes  then traveled 15 km . In this case ; 
if we take a time interval of 40 minutes:
or it contains part of the first 10 minutes, but then it does not include any part of the last 10 minutes, and in this case the distance traveled is less than 15km.
or it contains part of the last 10 minutes, but then it does not include any part of the first 10 minutes, and in this case the distance traveled is less than 15km.
or it is the "central" interval, and in this case the distance traveled is zero.
In any case, we can not find a time interval of 40 minutes during which he will have traveled 20km.
A: I don't know if this helps or hurts, but strictly speaking, there is no knowledge of the cyclist's speed at a specific point in time.  The cyclist may have a "magic rocket pack" and travel all 30km in 30 seconds, or he/she could have lesser rocket packs, or just generally bursts of energy, that result in an uneven rate of progress within the interval.
In all of these situations, case 2) is subsumed by case 1), and both are true.  
Now, the degenerate case as it happens, is an even rate of progress of 0.5km/min.  That is, there is no way to proceed for a full hour at any lesser rate and still achieve the distance of 30km.  (If so, then the cyclist must make it up with a burst at some other interval).  At the degenerate case, 0.5km/min is 5km in ten minutes, and also 20km in 40 minutes.
So generally, both assertions are true.
