) – f ( x ) > . We conclude that, whenever x , we have f x Remark If ′ ∀∈ a b , then for, x x a b [ , ] Î , such that x we have, f x > The proof is similar. Example . Find the values in the interval ( , ) of the mean value theorem satisfied by the function ( ) = for £ £ f ( ) and f ( ) = − .
Clearly f x ( ) is defined and differentiable in . Therefore, by the Mean Value Theorem, there exists a c ∈ ( , ) such that ′ f c ( ) = f = − That is, − c = − ⇒ Geometrical meaning Geometrically, the mean value theorem says the secant to the curve ( ) between x and x is parallel to a tangent line of the curve, at some point c a b ∈ ( , ) . Consequences of Lagrange’s Mean Value Theorem There are three important consequences of MVT for derivatives. ( ) To determine the monotonicity of the given function (Theorem .
) ( ) If ′ for all x in ( , ) a b , then f is constant on ( , ) a b . ( ) If ′ ′ g x ( ) for all x , then f x g x for some constant C . Tangent Secant f a f b ′ f c f b f a Fig. .