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

In summary, the conversation discusses using the Intermediate Value Theorem to prove two statements about continuous functions on closed intervals. For the first statement, it is shown that there exists an x in the interval where the function f(x) is equal to x. For the second statement, a suggestion is made to construct a new function g from f in order to make it easier to apply the theorem.
  • #1
LilTaru
81
0

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?!
 
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  • #2
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).
 
  • #3
Oh! Does that mean for (a) g(x) = f(x) - x? Or am I completely off track?!
 
  • #4
It's a good thought -- run with it and see where you get to.
 
  • #5
I am still very confused with how to form g(x)... it is still not making sense how to prove this question!
 
  • #6
LilTaru said:
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?
 

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

What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a mathematical theorem that states that if a continuous function has different values at two points, then it must have at least one value between those two points.

Why is the Intermediate Value Theorem important?

The Intermediate Value Theorem is important because it allows us to prove the existence of solutions to equations and problems that cannot be solved algebraically. It also helps us to understand the behavior of continuous functions.

What are the conditions for the Intermediate Value Theorem to hold?

The conditions for the Intermediate Value Theorem to hold are that the function must be continuous on a closed interval and have different values at the endpoints of that interval.

Can the Intermediate Value Theorem be applied to all functions?

No, the Intermediate Value Theorem can only be applied to continuous functions. Functions that have discontinuities, such as jumps or holes, do not satisfy the conditions for the theorem to hold.

How is the Intermediate Value Theorem used in real life?

The Intermediate Value Theorem is used in various fields of science and engineering to prove the existence of solutions to problems. For example, it can be used in physics to show that a moving object must pass through a certain point at a certain time, or in economics to prove the existence of a market equilibrium.

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