The Mean Value Theorem in Data Science: A Comprehensive Guide

Mean value theorem or Lagrange mean value theorem states that if f(x) is a continuous and differentiable function over an interval (a, b), then there exists at least one c in (a, b) that is equal to function average rate of change over the given interval. This article, will Mean Value Theorem, Rolles Theorem, and Cauchy Mean Value Theorem.

The Mean Value Theorem is one of the calculus’s most fundamental but most important theorems that can be used to identify the function’s behavior, whether it is increasing, decreasing, or constant at the given interval. This article will discuss the Lagrange Mean Value Theorem, Cauchy Mean Value Theorem, and Roll’s Theorem (a special case of the Lagrange Mean value Theorem).

Table of Content

• What is Mean Value Theorem
  • Rolle’s Theorem
  • Proof of Mean Value Theorem
  • Corollaries of Mean Value Theorem 
• Cauchy Mean Value Theorem

What is Mean Value Theorem

The Mean Value Theorem or Lagrange Mean Value Theorem states that:

If a function f(x) is continuous on [a, b] and differentiable on an (a, b), 

then there exists a point c in the interval (a, b) such that f’ (c) is equal to the function average rate of change over [a, b], i.e.,

From the above graph, we can easily depict that the tangent at any point c between [a, b] is parallel to the secant line over [a, b].

Now, what if f(a) = f(b), in the above MVT.

Yes, you depicted correctly, i.e., f’ (c) = 0. This is what Rolle’s Theorem is, i.e., Rolle’s Theorem is a special case of the Lagrange Mean Value Theorem.

Rolle’s Theorem

Rolle’s Theorem states that if f(x) is continuous over [a, b], differentiable over (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that

f’(c) = 0

Proof of Mean Value Theorem:

Let a line is passing through two points (a, f(a)), and (b, f(b)), then the equation of the line will be:

Consider a difference function h(x), that is equal to vertical difference between (x, f(x)) and (x,y) on that line, then

Corollaries of Mean Value Theorem

• If f’(x) < 0, for all x in (a,b), then 
  • f is strictly decreasing on (a,b).
• If f’(x) = 0, for all x in (a,b), then 
  • f is constant on (a,b).
• If f’(x) > 0, for all x in (a,b), 
  • then f is strictly increasing on (a,b).
• If f’(x) = g’(x), for all x in (a,b), then 
  • there exists a constant C, such that f(x) = g(x) for all x in (a,b).

Extended Mean Value Theorem

This is also called Cauchy Mean Value Theorem. It states that if f, and g are continuous over [a, b], differentiable over (a, b), and g’ (x) is not equal to zero for any point in (a, b), then there exists at least one point c in (a, b), such that:

It can be easily derived from the Lagrange Mean Value theorem:

For any continuous and differentiable functions f, and g over (a, b), there exists at least one point c in (a, b) such that:

Application of Mean Value Theorem

• Leibnitz Rule
• L’Hopitals Rule
• Symmetry of Second Derivative

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