What is the Proof for Rolle's Theorem Using the Intermediate Value Theorem?

[rogain]

Homework Statement

A function is continuous/differentiable on the interval [0,3]. I have been given that f(0)=1, f(1)=2, f(3)=2.
I need to prove that there exists a c within the interval [0,3] such that f(c) = c.

Homework Equations

The Attempt at a Solution

f(1)=f(3)=2. From Rolle's theorem, there exists a c within [1,3] such that f`(c)=0. From this I am able to show that at some point c within [1,3] the function must be constant i.e f(c)=k.
I am not sure how to show that the constant is equal to the point i.e k=c.

 

[HallsofIvy]

The problem does not ask you about the derivative of f so Rolle's theorem is unnecessary. Also it makes no sense to say that a function is constant at a point! Every function has a single value at every point of its domain.

Define g(x)= f(x)- x. g(0)= 1- 0= 0, g(1)= 2- 1= 1, and g(3)= 2- 3= -1. That is, g(1)> 0 and g(3)< 0. What does that tell you?


(Much of the information given in this problem is unnecessary! The function does not have to be differentiable, just continuous. The information about f(0)= 1 is irrelevant. And, in fact, the desired point must be in the interval [1, 3].)

 

[rogain]

Define g(x)= f(x)- x. g(0)= 1- 0= 0, g(1)= 2- 1= 1, and g(3)= 2- 3= -1. That is, g(1)> 0 and g(3)< 0. What does that tell you?

Thanks for the response. This helped a lot.

 

[Dickfore]

No, they're invoking The Intermediate Value Theorem.