How Does the Intermediate Value Theorem Apply to Finding Fixed Points?

[Latios1314]

Have a couple of questions that I've been stuck with for some time. Would greatly appreciate the help. No idea how I should approach these questions and how do I start.

6. Suppose that a function f is continuous on the closed interval [0,1]and that 0≤f(x)≤1 for every x in [0,1].Show that there must exist a number c in [0,1] such that f(c)=c(c is called a fixed point of f).

(7) Is it true that if you stretch a rubber band by moving one end to the right and the other end to the left, some point of the band will end up in its original position? Give reasons for your answer.

(8) Is there any reason to believe that there is always a pair of antipodal (diametrically opposite) points on Earth’s equator where the temperatures are the same? Explain.

 

[gb7nash]

6. Suppose that a function f is continuous on the closed interval [0,1]and that 0≤f(x)≤1 for every x in [0,1].Show that there must exist a number c in [0,1] such that f(c)=c(c is called a fixed point of f).

Consider g(x) = f(x) - x. What property does this function have? Try to think what else you could do with g(x).

 

[Latios1314]

From where do I derive the equation g(x)=f(x)-x?

 

[Latios1314]

I reckon that I should perhaps use composite functions to do this but I have no idea from where can i get the function g(x) from.

 

[Latios1314]

Managed to get both questions 6 and 7 down.

Not sure whether my working for 7 is correct though.

Let x1 be the new position of the left end of the elongated rubber band.
let d(x) be the displacement from the original position.

d(x) = x - x1 <0

Let x2 be the new position of the right end of the elongated rubber band.
d(x) = x2 - x >0

Therefore by Intermediate Value Theorem, d(x) = 0 must exist. Hnece some part of the elongated rubber band must be at the oringal position.

 

[Latios1314]

how do i go about question 8 though?

I don't really get the question. Know that i must make use of intermediate value theorem somehow but i have no idea how I'm supposed to do it.