Mean Value Theorem Statement

This special section will kick out your fear of a very important topic of calculus in Mathematics which is useful for both differential and integral calculus. Forgoing this topic, one should have basic knowledge of limit, continuity, and differentiability as it deals with these concepts. Rolle’s Theorem is very basic. Further, we broadly discuss its terms in Mean Value Theorem with some special conditions. It is important to find the roots of a polynomial, Leibniz’s theorem, L’Hospital’s rule, and the symmetry of second-order derivatives. The Mean Value Theorem is also known as Lagrange’s Mean Value Theorem.  

Rolle’s Theorem

If:

(i)  f(x) is continuous in the closed interval [a, b]

(ii) f(x) is differentiable in the open interval (a,b) i.e f'(x) exists for every value of x in interval (a,b) and

(iii) f(a)=f(b) 

Then, at least one value, say c of x in (a,b), such that f'(c) = 0.

All three conditions should be satisfied for applying Rolle’s Theorem. 

Lagrange’s Mean Value Theorem

There are two forms of this theorem which are mentioned below:

First form: (i) if f(x) is continuous in the closed interval [a, b] and

(ii) f(x) is differentiable in the open interval (a,b) i.e f'(x) exists for every value of x in interval (a,b), then there exists at least one value c of x in (a,b) such that 

f'(c) = f(b) – f(a) / b-a

Second form: let us consider b= a+h, where h is a small value, then since a<c

Then the Mean Value Theorem  can be represented as: if, 

(i) f(x) is continuous in the closed interval  [a,a+h] and

(ii) f(x) is differentiable in the open interval (a,a+h) i.e f'(x) exists for every value of x in interval (a,a+h), then there is at least one number θ such that 

f(a+h)=f(a) + f'(a+θh) 

Geometrical interpretation of Mean Value Theorem

This theorem asserts that if a curve has a tangent at each of its points, there is at least one point c on this curve, the tangent at which is parallel to its chord. 

That’s all for the statement of Lagrange’s Mean Value Theorem. 

Cauchy’s Mean Value Theorem

This theorem is applicable when two functions are present. Let us consider the functions as f(x) and g(x). 

If, (i) f(x) and g(x) be continuous in the closed interval [a,b] 

(ii) f'(x) and g'(x) both exist in (a,b) i.e both are functions should be differentiable in the interval (a,b) and

(iii) g'(x) <>0  for any value if x in (a,b) interval, then there exit at least one value of c in (a,b) such that

 f(b) – f(a) /g(b) -g(a) = f'(c) /g'(c) 

It is the required formula for Cauchy’s Mean Value Theorem 

This theorem is nothing but a generalisation of LAGRANGE’S Mean Value Theorem.

Conclusion

After having the whole idea of this Mean Value Theorem, we can conclude that it is an important topic for Calculus. It states that if a function is continuous in a given closed interval and at the same time it is also differentiable in the same open interval, then there exists at least one point between the given interval at which the tangent is equal to the slope of the secant line between the given interval. Sometimes it is also stated as MVT. Before going through this topic, one must know limits, continuity, and differentiability. 

That’s all for this topic; I hope it’s going to help you all greatly.