Mean Value Theorems

It is a very important topic for jee students because every year ,in advance and mains ,there are 2-3 questions based on 2 concepts 

1. Rolls theorem:Let f">ff be a continuous function over the closed interval [a,b]">[a,b][a,b] and differentiable over the open interval (a,b)">(a,b)(a,b) such that f(a)=f(b)">f(a)=f(b)f(a)=f(b). There then exists at least one c∈(a,b)">c∈(a,b)c∈(a,b) such that f′(c)=0">f′(c)=0f′(c)=0.

2. Lagranges theorem:

Lagrange's mean value theorem is also known as the mean value theorem or MVT or LMVT. It states that if a function f (x) is a continuous in a close interval [a, b] where (a≤x ≤b) and differentiable in open interval [a, b] where (a < x< b), then there is at least one point x = c on this interval, given as

f(b) - f (a) = f' (c) (b-a)

The above theorem is called first mean value theorem. It enables to express the increment of a function in the given interval through the derivative value at an intermediate point of the segment.