Rolle’s Theorem and Lagrange’s Mean Value Theorem

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Rolle’s Theorem and Lagrange’s Mean Value Theorem in Detail

In calculus mathematics, Rolle's Theorem and Lagrange's mean value theorem constitute a significant portion of the syllabus that students need to learn. These theorems also conclude the truth of converse and when satisfy three fundamental conditions.

Typically, Rolle’s Theorem is an extension or special case of mean value theorem and therefore is interrelated. However, if you want to know more about its application and formula, refer to the next segments that elucidate these concepts for you.

Thus, let’s first start with the definition of Lagrange’s Mean value theorem.


What is Lagrange's Mean Value Theorem?

This theorem says that in a particular curve if f is a function and a and b are two endpoints, there has to be a point where the tangent curve of the slope is equal to the slope of secants through two endpoints.

Here, the conditions of this theorem would be:

  1. a and b are two variables in the continuous closed interval [a,b].

  2. They are differentiable in open intervals, (a, b).

  3. f (a) is not equal to f(b).

If these conditions satisfy, then there would be point c between a and b where a < c < b.

Ultimately, the statement of Lagrange's mean value theorem concludes that the function f that has a horizontal tangent in the interval.

For more insight, you have to prove this theorem with step by step expressions.


Lagrange's Mean Value Theorem Proof

Following is the proof of Lagrange's mean value theorem:

Consider h(x) = f(x) - [{f(b) - f(a)}/(b - a)](x - a), where h(a) = h(b) = f(a)

Rolle’s theorem c is in (a, b) such that h’(c) = 0.

Or,

f’(c) - [{f(b) - f(a)}/(b - a)] = 0

f(b) - f(a) = f’(c) (b - a)

This mean value theorem is also known as the first mean value theorem. Moreover, this one is applicable in mathematics as well as in computational mathematics, etc.

In maths paper, you can expect questions like “state and prove Lagrange's mean value theorem”. Thus, you need to be prepared to explain the theorem as mentioned above.

Moreover, if you want to score better in calculus, you also have to provide a graphical interpretation of Rolle's Theorem and Lagrange's mean value theorem. In this case, you can refer to the following segment.


Graphical Interpretation of Lagrange's Mean Value Theorem

Following are three variants where you can state Lagrange's mean value theorem.

In every instance, f is a function, and a and b are in interval and c is the point in between the interval. If you draw a tangent f’(c), you will see that the tangent is horizontal and f(a) and f(b) are equal.

You can use the following diagram to substantiate this theorem.


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Based on this graph, you can easily derive Lagrange's mean value theorem formula. However, besides explaining this theorem, you also have to know how to connect to Rolle’s Theorem.


How Rolle's Theorem and Lagrange's Mean Value Theorem Connected?

Rolle’s theorem is the foundation of the mean value theorem. If a function fulfils first 2 conditions of mean value theorem, then there would be c at (a, b) such that f’(c) = 0, where f(a) = f(b).

Like the Mean value theorem, Rolle’s value theorem also has graphical representation that is essential for students to learn.


Graphical Representation of Rolle’s Theorem

You can use any of the following diagrams to prove this theorem.


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Thus, after knowing Rolle's Theorem and Lagrange's mean value theorem, you can solve several problems on that same with ease. However, you need to practise these factors often so that you can remember them and fetch good marks in the maths examination.


Moreover, these concepts are also asked in almost all entrance exams that you need to solve within a few minutes. Thus, thorough practise and solving mathematical problems are of the utmost importance to get a firm grip on these topics. To better your performance on this topic, you can opt for assistance from leading e-learning platforms like Vedantu. With notes from subject experts, paired with live online classes, and doubt clearing sessions, you can get a better grip on this relatively difficult topic.

FAQ (Frequently Asked Questions)

1. What are the Major Differences Between Lagrange’s Mean Value Theorem and Rolle’s Theorem?

Ans. Let’s assume, f is a Function and a and b are two variables in closed intervals, continuous, differentiable on an open interval. Then, a tangent is parallel to joining sections, (a, f(b)) and (b, f(b)) at c point in the mean theorem. On the other hand, in Rolle’s Theorem, a tangent is parallel to x-axis at c point.


While the main concept is Lagrange’s mean value theorem, Rolle’s theorem is a special variant of it or extension of the primary concept. This difference helps students to learn about both these concepts well.

2. Where can you not Use Rolle’s Theorem?

Ans. Even if a function f continuous on closed interval, differentiates in an open interval and f(a)= f(b), then you can apply Rolle’s theorem to solve a problem. However, these are some exceptions as well that you need to understand where you cannot use this theorem.


For instance, f(x) ={x} (where {x} is a fractional part function) on [0,1], a closed interval . You can derive from the function that on this open interval (0,1) derivative equals to 1, everywhere. In such a case, Rolle’s theorem does not apply because f(x), the function has a discontinuity or break at x = 1 which implies the function is not continuous on all places of [0,1], on the closed interval.

3. What does Rolle’s Theorem Show?

Ans. Rolle’s theorem states that in case of a constant function, the graph of it would be a horizontal line segment. Simultaneously, it also fulfils all conditions of Rolle ’s Theorem as the derivative is 0 everywhere.


However, you need to remember that this theorem guarantees a minimum one point if not multiple points. Yet, to answer this kind of question, you also have to provide a graphical representation of this theorem to substantiate your answer. Only then, you can secure full marks in this part.