Rolle’s+Theorem

=__Rolle's Theorem__=

Rolle's Theorem is an extension of the Mean Value Theorem. Rolle's Theorem applies the Mean Value Theorem to situations in which the two endpoints have the same value.

If the function f(x) is continuous and differentiable on the interval (a,b) and f(a)=f(b), then there exists some point c such that f '(c)=0.
 * __Rolle's Theorem__**

This can be derived very simply from the Mean Value Theorem by setting f(a)=f(b): Rolle's Theorem states that a function will have a horizontal tangent line somewhere within (a,b) if f(a)=f(b):

Show that f(x)=4x 5 +x 3 +7x-2 has exactly one real root. (Problem and solution adapted from [])
 * Example One **

Solution: As this is a polynomial, f(x) will be continuous and differentiable for all values of x, so the Intermediate Value Theorem and Rolle's Theorem will hold. Note that f(0)=(-2) and f(1)=10. By the Intermediate Value Theorem, there will be a root on the interval 0<x<1.

If there were two or more roots, then there would be two or more places where f(a)=f(b)=0. So we can apply Rolle's Theorem to find where f '(x)=0: f '(x)=20x 4 +3x 2 +7=0 As the even-powered terms will always be greater than or equal to zero, the minimum value of this function is 7. As 7 does not equal 0, Rolle's Theorem does not apply, proving that there is only one location where f(x)=0.