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

[skyturnred]

Homework Statement

I am trying to prove that f(x)=x$^{4}$-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.

 

[jgens]

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?

 

[skyturnred]

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).

 

[jgens]

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.

 

[skyturnred]

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?