Prove that f(x)=x^4 - x - 1 has exactly one root

  • Thread starter skyturnred
  • Start date
  • Tags
    Root
In summary: Yes, this is correct. If the function had only one root in the interval, but it still changed direction multiple times, then Rolle's Theorem would not be able to prove that it has only one root. It would instead be able to prove that it has LESS than two roots.
  • #1
skyturnred
118
0

Homework Statement



I am trying to prove that f(x)=x[itex]^{4}[/itex]-x-1 has exactly one root on [1,2].

Homework Equations





The Attempt at a Solution



Step one, by intermediate value theorem I proved that there is at least one root on [1,2]. But I don't know how to go about proving there is only that single root. We were taught how to do this with Rolle's Theorem, but since this does not satisfy the conditions for Rolle's Theorem, I can't use it.

I think that I must somehow use the mean value theorem.. But I don't know how.
 
Physics news on Phys.org
  • #2
Suppose f had two roots in [1,2]. Then by Rolle's Theorem, its derivative would be zero somewhere in (1,2). Can you use this to finish the proof?
 
  • #3
jgens said:
Suppose f had two roots in [1,2]. Then by Rolle's Theorem, its derivative would be zero somewhere in (1,2). Can you use this to finish the proof?

Yes, thanks. The derivative is only zero at somewhere around 0.6 which is not in the interval, therefore it must be increasing or decreasing ONLY throughout [1,2]. Since it is a continuous function it can have a maximum of one root. This proves, of course, that there is EXACTLY one root because of intermediate value theorem.

But is it correct to say that this is done by Rolle's Theorem? Because the function does not satisfy the conditions for Rolle's Theorem, in other words, f(1) does not equal f(2).
 
  • #4
We are using Rolle's Theorem to derive a contradiction. If the function did have two roots, then Rolle's Theorem says the derivative of the function must be zero somewhere between those roots. Since this is not the case, we have at most one root. So yes, it is correct to say that this is done by Rolle's Theorem.
 
  • #5
jgens said:
We are using Rolle's Theorem to derive a contradiction. If the function did have two roots, then Rolle's Theorem says the derivative of the function must be zero somewhere between those roots. Since this is not the case, we have at most one root. So yes, it is correct to say that this is done by Rolle's Theorem.

Ok, thanks! That makes sense! Just so I can understand properly though, what if with the interval of [1,2] the function actually had only one root, but still changed direction multiple times? Based on this, is it correct to say that Rolles Theorem can only be used to prove by contradiction that something has LESS than 2 roots, and not AT LEAST two roots?
 

1. What does it mean for a function to have exactly one root?

A root of a function is a value of x that makes the function equal to zero. Having exactly one root means that there is only one value of x that satisfies the equation f(x)=0.

2. How can you prove that a function has exactly one root?

One way to prove that a function has exactly one root is by using the Intermediate Value Theorem. This theorem states that if a continuous function takes on positive and negative values at two points, then it must also take on the value of zero at some point in between.

3. How do you find the root of a function?

To find the root of a function, you can set the function equal to zero and solve for the variable x. In the case of f(x)=x^4-x-1, you can use algebraic methods or a graphing calculator to find the approximate value of the root.

4. What is the significance of a function having exactly one root?

A function having exactly one root means that it crosses the x-axis at only one point, which can be useful in many applications. It also means that the function can be easily graphed and analyzed.

5. Can a function have more than one root?

Yes, a function can have more than one root. In fact, some functions may have an infinite number of roots. However, in the case of f(x)=x^4-x-1, we can use the Intermediate Value Theorem to prove that there is only one root.

Similar threads

  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
16
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
641
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
119
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
956
  • Calculus and Beyond Homework Help
Replies
2
Views
608
Back
Top