. . Lagrange’s Mean Value Theorem Theorem . Let f x ( ) be continuous in a closed interval [ , ] a b and differentiable in the open interval ( , ) a b (where f ( a ), f ( b ) are not necessarily equal).
Then there exist at least one point a b ∈ ( , ) such that, ′ f c ( ) = f b f a ... ( ) Remark If f a f b then Lagrange’s Mean Value Theorem gives the Rolle’s theorem. It is also known as rotated Rolle’s Theorem . Remark A physical meaning of the above theorem is the number f b f a can be thought of as the average rate of change in f x ( ) over ( , ) a b and ′ f c ( ) as an instantaneous change.
A geometrical meaning of the Lagrange’s mean value theorem is that the instantaneous rate of change at some interior point is equal to the average rate of change over the entire interval. This is illustrated as follows : f a f b ′ f b f a Fig. . - - Applications of Differential Calculus If a car accelerating from zero takes just seconds to travel m, its average velocity for the second interval is m/s.
The Mean Value Theorem says that at some point during the travel the speedometer must read exactly km/h which is equal to m/s. Theorem . If f x ( ) is continuous in closed interval [ , ] a b and differentiable in open interval ( , ) a b and if ′ > ∀∈ a b , then for, x x a b [ , ] , such that x we have, f x Proof By the mean value theorem, there exists a c x x a b ⊂ ( , such that, = ′ f c ( ) Since ′ f c ( ) > , and x > we have f ( x