How to use Rolle's Theorem to prove exactly ONE REAL ROOT

[Khayyam89]

Homework Statement

Show that the equation 2x-1-sin(x) = 0 has exactly one real root.

Homework Equations

The Attempt at a Solution

I first used the Intermediate Value Theorem to prove that there exists at least one c such that f '(c)=0. The next step requires Rolle's Theorem to prove that there is EXACTLY ONE REAL ROOT, but I have no idea how to proceed at this point.

 

[Dick]

I think you mean that you used the IVT to show there is a least one value of c such that f(c)=0. Not f'(c). f'(x) is never zero. Can you show that? If so what does Rolles theorem tell you if f(x)=0 had multiple roots.