. . Rolle’s Theorem Theorem . (Rolle’s Theorem) Let f x ( ) be continuous on a closed interval [ , ] a b and differentiable on the open interval ( , ) a b If f a f b , then there is at least one point c a b ∈ ( , ) where ′ c ( ) .
Geometrically this means that if the tangent is moving along the curve starting at x towards as in Fig . then there exists a c a b ∈ ( , ) at which the tangent is parallel to the x -axis. Example . Compute the value of ' ' c satisfied by the Rolle’s theorem for the function ) , [ , ] .
Observe that, f ( ), is continuous in the interval [ , ] and is differentiable in . Now, ′ f ( ) = )( Fig. . ′ f c ( ) - - Therefore, ′ f c ( ) = gives c = , , and which ⇒ c = ∈ ( , ) .
Example . Find the value in the interval satisfied by the Rolle's theorem for the function x x ∈ . We have, f x ( ) is continuous in and differentiable in with f = ( ) . By the Rolle’s theorem there must exist a value c ∈ such that, ′ = − ⇒ gives c = ± , .
As ∈ , we choose c = . Example . Compute the value of ' ' c satisfied by Rolle’s theorem for the function f x in the interval [ , ] . Observe that, f and f x ( ) is continuous in the interval [ , ] and differentiable