Rolle's Theorem and Lagrange's Mean Value Theorem

Rolle’s Theorem is a special case of the mean value of theorem which satisfies certain conditions. Whereas Lagrange’s mean value theorem is the mean value theorem itself or also called first mean value theorem. Here in this article, we will learn both the theorems. By mean we understand the average of the given values. But in the case of integrals, the process of finding the mean value of two different functions is different.

Lagrange’s Mean Value Theorem

If a function f  is defined on the closed interval [a,b] satisfying the following conditions –

i) The function f is continuous on the closed interval [a, b]

ii)The function f  is differentiable on the open interval (a, b)

Then there exists a value  x =  c in such a way that

f'(c) = [f(b) – f(a)]/(b-a)

This theorem is also known as the first mean value theorem or Lagrange’s mean value theorem.

Geometrical Interpretation of Lagrange’s Mean Value Theorem

In the given graph the curve y = f(x)  is continuous from x = a and x = b and differentiable within the closed interval [a,b] then according to Lagrange’s mean value theorem,for any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c.

\(~~~~~~~~~~\) \( f'(c) \) = \( \frac {f(b) – f(a) }{b -a} \)

Rolle’s Theorem

A special case of Lagrange’s mean value theorem is Rolle ’s Theorem which states that:

If a function f  is defined in the closed interval [a,b] in such a way that it satisfies the following conditions.

i) The function f is continuous on the closed interval [a, b]

ii)The function  f is differentiable on the open interval (a, b)

iii) Now if f (a) = f (b) , then there exists at least one value of x, let us assume this value to be c, which lies between a and b i.e. (a < c < b )  in such a way that f‘(c) = 0 .

Precisely, if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) then there exists a point x = c  in (a, b) such that f'(c) = 0

Geometric interpretation of Rolle’s Theorem

In the given graph, the curve y = f(x)  is continuous between x =a  and x = b and at every point within the interval it is possible to draw a tangent and ordinates corresponding to the abscissa and are equal then there exists at least one tangent to the curve which is parallel to the x-axis.

Algebraically, this theorem tells us that if f (x) is representing a polynomial function in x and the two roots of the equation f(x) = 0 are x =a and x = b, then there exists at least one root of the equation f‘(x) = 0  lying between the values.