Prove: Mean Value Theorem & Rolle's Theorem | At Most 1 Fixed Point

In summary: This is a valid argument, but it doesn't use the Mean Value Theorem or Rolle's Theorem.In summary, if f is a differentiable function with a constant derivative of 1 for all x, then it can have at most one fixed point. This is because if f(a)=a, then for k=0 the function has no fixed points and for k≠0 the function has every point as a fixed point, resulting in a contradiction. However, this argument does not utilize Rolle's Theorem or the Mean Value Theorem.
  • #1
DeltaIceman
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Homework Statement


A number a is called a fixed point if f(a)=a. Prove that if f is a differentiable function with f'(x)=1 for all x then f has at most one fixed point.


Homework Equations


In class we have been using Rolle's Theorem and the Mean Value Theorem.


The Attempt at a Solution


In all honest I wasn't sure where to start but this is what I've come up with so far. Knowing that the slope or f'(x)=1 then the original function must have been something like f(x)= x + k. Considering k as a constant that could exist or could not. Then the function either has no fixed point. Or every point of the function is fixed. Therefore giving us a contradiction in the statement. Meaning that this statement cannot be possible. We worked a couple of these in class and I didn't really know how to approach this problem. What I did kinda makes sense to me although it doesn't seem like this should be the answer. Any help would be appreciated thanks!
 
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  • #2
Let [itex]x=a[/itex] be a fixed point of [itex]f[/itex]. Then by the definition of "fixed point", [itex]f(a)=a+k=a[/itex]. Consider 2 cases: [itex]k=0[/itex] and [itex]k\neq 0[/itex].
 
  • #3
Hold on a second...The proposition in the problem statement is false. Let [itex]f(x)=x[/itex]. Then [itex]f'(x)=1[/itex] for all [itex]x[/itex], and every point is a fixed point!
 
  • #4
Agreed.
 

Related to Prove: Mean Value Theorem & Rolle's Theorem | At Most 1 Fixed Point

1. What is the Mean Value Theorem?

The Mean Value Theorem states that if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists at least one point c in (a,b) where the slope of the tangent line at c is equal to the average rate of change of the function over the interval [a,b].

2. How is the Mean Value Theorem related to Rolle's Theorem?

Rolle's Theorem is a special case of the Mean Value Theorem, where the average rate of change over the interval [a,b] is equal to zero. This means that there exists at least one point c in (a,b) where the derivative of the function is equal to zero, or in other words, where the function has a horizontal tangent line.

3. What is a fixed point in the context of these theorems?

A fixed point is a point on a function where the input and output values are equal. In other words, a fixed point is a solution to the equation f(x) = x. The Mean Value Theorem and Rolle's Theorem both involve finding points on a function where the derivative is equal to zero, which can also be considered fixed points.

4. Can there be more than one fixed point on a function?

Yes, there can be multiple fixed points on a function. However, the Mean Value Theorem and Rolle's Theorem both state that there can be at most one fixed point on a function within a given interval.

5. How are these theorems used in real-world applications?

The Mean Value Theorem and Rolle's Theorem are used to prove the existence of solutions to equations and problems in various fields, such as physics, engineering, and economics. For example, the Mean Value Theorem is used in calculus to prove the Fundamental Theorem of Calculus, which is essential for many applications in physics and engineering.

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