Consider the functions,
s(t)and r(t) which describes the position of the man when he is running up and running down, respectively.
Let,
f(t)=s(t)−r(t) and consider the interval [0,10].
Note that s(t) and r(t) are continuous since they are position functions. Thus, f(t) is continuous as well.
Now, note that
f(0)=s(0)−r(0)<0
since the initial position when running down (which is at the top of the mountain) is higher than the initial position when running up (which is at the foot of the mountain). Moreover,
f(10)=s(10)−r(10)>0
since at this time, the final position when running down is at the foot of the mountain which is less than the final position when running up (which is halfway point up the mountain).
Observe that 0 is between f(0) and f(10).
Hence, by the Intermediate Value Theorem, there
exists a point x∈(0,10) such that f(x)=0.
This means that
f(x)=0
⇒s(x)−r(x)=0
⇒s(x)=r(x)
Therefore, at time t=x, the man is at the same point when he is running up on a Saturday and running down on a Sunday.
This is from the intermediate value theorem :
Assume that we have a continuous function f(x) on some interval, say [a,b].
As defined in mathematics, the Intermediate Value Theorem states that if we have a value c which is between f(a) and f(b), then we can find a point between the interval, say x∈(a,b), such that f(x)=c.
Disclaimer : The complete question is attached.
Learn more about intermediate value theorem at :
https://brainly.com/question/4940880
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