Using the Intermediate Value Theorem to Solve for x in Continuous Functions

[LilTaru]

Homework Statement

a) Suppose that f(x) is a continuous function on [0, 1] and 0 <= f(x) <= 1 for all x in [0, 1]. Show that there is an x in [0, 1] where f(x) = x.

b) Suppose that f(x) is a continuous function on [0, 2] with f(0) = f(2). Show that there is an x in [0, 1] such that f(x) = f(x + 1).

Homework Equations

The Attempt at a Solution

I assume I have to use the Intermediate Value Theorem, but I have no idea how to use it! For (a) I thought 0 <= f(x) <= 1 means since f(x) is between f(a) and f(b) then there exists a c or x in this question so that f(x) = x, but I have no idea! And for (b)... not a clue! Please help?!

 

[ystael]

Try to construct a new function g from f, in such a way that answering the question means comparing g to a constant (so that it's easier to apply the intermediate value theorem).

 

[LilTaru]

Oh! Does that mean for (a) g(x) = f(x) - x? Or am I completely off track?!

 

[ystael]

It's a good thought -- run with it and see where you get to.

 

[LilTaru]

I am still very confused with how to form g(x)... it is still not making sense how to prove this question!

 

[Dick]

I am still very confused with how to form g(x)... it is still not making sense how to prove this question!

You already formed a good g(x) by setting g(x)=f(x)-x. If g(0)=0 then f(0)=0 and you are done. Otherwise g(0) is positive, right? What happens if g(1)=0? Suppose g(1) is not zero. What sign is it?