CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 51, 2016 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Tichun Wang, Hongyang Zhang, Lei Tian 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-43-3; ISSN 2283-9216 

Proof of Lagrange Mean Value Theorem and its Application in 
Text Design 

Jianhua Li 
Mathematical Science College, Luoyang Normal University, Luoyang, 471000, P.R.China 
ljhly2010@163.com 

At present, there are a lot of papers on Lagrange mean value theorem proving method, the paper On the 
application of the theorem is not in a few, but text designs from the perspective of curriculum explore proving a 
theorem and its application to the design of a text are rare. This paper first analyzes the objectives, tasks, 
methods, and then focuses on the teaching strategies and processes, Finally, it comes to the reflection, thus 
forming a complete, detailed and full text design on the proof of Lagrange mean value theorem and its 
application. In this paper, through study of the teaching content of the Lagrange mean value theorem, 
optimizing the teaching design, adopting it in the classroom teaching, the paper gives the classroom teaching 
practice in the specific practices, thereby improves the effect of classroom teaching, and it has a strong 
guiding significance in theory and practice. 

1. Introduction 

The differential mean value theorem is the theoretical basis of the application of the derivative, which is the 
bridge between the function and its derivative. It is an important tool to study the function overall by using 
derivative of the locality, and forms a relation between the increment of a function on an interval and the 
derivative of the function at a specific point of the interval. As a result, differential mean value theorem has 
important theoretical significance and application value. 

2. Main text 

2.1 Task analysis 

"Lagrange mean value theorem and its application" belongs to the first section of the third chapter numbering 
in the first volume of "Higher mathematics" (6th Edition) (Department of mathematics, Tongji University, higher 
education press), the third chapter is the application of the differential mean value theorem and the derivative. 
The content of this chapter is the continuation of the last chapter. It mainly analyzes and studies the character 
of function and its graph and various forms by using derivative and differential (Farid, 2015). The theoretical 
basis of all of this is the differential mean value theorem, which plays an important role in the differential 
calculus, is also the content of the first section (Zhang, 2015). The differential mean value theorem includes 
Rolle theorem, Lagrange mean value theorem, Cauchy mean value theorem and Taylor mean value theorem. 
In the process of analysis and demonstration, the mean value theorem is widely used. The teaching task of 
this course is to study Lagrange mean value theorem and the application of theorem in equality and inequality 
(Mortici, 2011). 

2.2. Objectives and basic requirements 

Knowledge and Skill Goals: (1) through the study of this course, students should understand and master the 
conditions and conclusions of Lagrange mean value theorem; (2) to make the students understand Lagrange 
mean value theorem and the geometric meaning of Lagrange mean value theorem, and the connection and 
difference between Rolle theorem and Lagrange mean value theorem; (3) through the study of this course, 
students should learn how to use the mean value theorem to prove the equality and inequality. 
Ability Training Goal: (1) to improve students’ analysis ability and logical thinking ability of proving 
mathematical proposition with the cycle from conclusion to the conditions and the conclusion (Yi, 2015). The 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651055

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Li J.H., 2016, Proof of lagrange mean value theorem and its application in text design, Chemical Engineering 
Transactions, 51, 325-330  DOI:10.3303/CET1651055   

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following section is to prove the conclusion of Lagrange mean value theorem: There is at least one point f(b)-
(ab) Inside (a,b). In the process of making the equation  

))(()()(
'

abfafbf                                                                           (1) 

Established, Starting from the conclusion f(b)-f(a)=f’()(b-a), through the analysis of this equation, the auxiliary 
function is constructed, and the condition that the function f(x) satisfies in the theorem is used to prove the 
Lagrange mean value theorem by Rolle theorem.                       
(2) To develop students’ ability to use the construction of auxiliary function to prove the mathematical 
proposition while proving Lagrange mean value theorem (Heydari, 2013), (Tan, 2014) a relation between the 
Lagrange mean value theorem and Rolle's Theorem is found from the geometric meaning. In order to prove 
the Lagrange mean value theorem with the Rolle theorem, Starting from the equation f(b)-f(a)=f’()(b-a) itself
，through deformation of the equation, and by making use of the property of the function and the derivative 
function, the auxiliary function  

x
ab

afbf
xfxF






)()(
)()(                                                                    (2) 

As the auxiliary function F(x) satisfies the condition of Rolle's Theorem, we prove the Lagrange mean value 
theorem by using the Rolle theorem (Korobkov, 2001), (Elbrous, 2008) in this process, students are taught to 
prove the mathematical proposition with auxiliary function. 
(3) To cultivate students' creative thinking ability. The college mathematics teaching should not only let the 
students know the existing research methods and conclusions, but also need to cultivate students' ability to 
discover, think about and solve new problems. In the course of teaching, I should take the following innovative 
try: In the process of proving the Lagrange mean value theorem, because the function f(x) does not satisfy the 
condition of Rolle theorem, I will start with the conclusion of the Lagrange mean value theorem, and guide the 
students to analyze the equation 
f(b)-f(a)=f’()(b-a) with the help of the nature of the function and the derivative function, the auxiliary function 
F(x)= f(x)-((f(b)-f(a))/(b-a)) is structured, and Lagrange mean value theorem is proved by Rolle theorem. The 
teaching material starts from the geometric meaning of the Lagrange mean value theorem, combined with the 
graphics 
 

 
 

Figure 1: Geometric sketch of Lagrange mean value theorem 

The auxiliary function 

)](
)()(

)([)()( ax
ab

afbf
afxfxF 




                                                          (3) 

is constructed by analyzing the characteristic of the length function of the line segment NM. Similarly, to the 
function F(x), the Lagrange mean value theorem is proved by the Rolle theorem (Tong, 2000). In the process, 
I guide the students to compare the two kinds of auxiliary functions and detect the characteristics and the 
differences between them (Zabrdjko, 1996), and analyze the constructing context and perspective of the two 
auxiliary functions. By contrast, make the students experience different (Lupu, 2009), multi-angle method of 
constructing auxiliary function. Meanwhile, by inspiring students to think whether different auxiliary functions 
can be constructed to prove theorems, I guide the students to find the problem, propose the question, analyze 
question, and solve the problem, cultivating students’ awareness of innovation and innovative thinking ability. 

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(4) To cultivate students’ ability of comprehensive analysis of the problem and the comprehensive use of 
Through the proof of the theorem and the proof of two examples on the equality and inequality by the theorem, 
knowledge, and with the help of the assignments, the students are supposed to grasp the following two kinds 
of abilities of analyzing problems and the comprehensive use of knowledge.  

2.3 Teaching Focus and Difficulty 

Teaching Focus: Lagrange mean value theorem 
Teaching Difficulty: the Construction of auxiliary function, the application of Lagrange mean value theorem  

2.4 Teaching Strategies  

Review the Old and Learn the New, Put forward the Question, Introduce New Courses: Looking back on the 
learned knowledge:  
Review the Old: Rolle theorem  If the function f(x) is satisfied: (1) be continuous on the closed interval [a,b], (2) 
be derivable in the open interval (a,b), (3) be equal to the value of the function at the point of the interval, that 
is f(a)=f(b).Then there is at least one point (ab), in the interior (a,b) so that the derivative of the function 
f(x) at that point is equal to zero, that is f’()=0 
Ask the question: if the third condition of Rolle's theorem is moved, what conclusion can we get (Comparative 
inspiration, to be discussed) 
Introduce new courses: Lagrange mean value theorem and its application 
Heuristic Induction, Theorem Proving: In each step of the class, including the discussion of the geometric 
significance of theorem, the construction of the auxiliary function from the analysis of the conclusion of 
theorem, the proof of the Lagrange mean value theorem by the Rolle theorem, the analysis and proof of the 
two examples of equality and inequality (Liu, 2015), I duly inspired and guided the students to think actively, 
participate in the classroom, and seek the functions that meet the conditions and the corresponding interval, 
thereby successfully applied theorem to prove the problems related. 
Method Training, Migration and Expansion: Teacher-oriented class emphasizes the one-way knowledge 
transference of the teacher and the memorizing work on the students, is the teachers’ "autocracy", "chalk and 
talk". While the students-oriented teaching reform attaches importance to comprehensive assistance from the 
teacher to the students and should provide students opportunities for autonomous learning on the students’ 
part (Niu, 2015), and give students time and tasks for mental work in order to facilitate students to master the 
learning method and use of migration in the new situation. After the proof of the problems of the equality in 
example 1 with Lagrange mean value theorem (Zhu, 2015), the problem is proposed that whether the theorem 
also has application in other aspects. The inequality proof problem of 2 is drawn off. That is to prove when 
x0, 

xx
x

x



)1ln(

1
                                                                                (4) 

(This question has a certain degree of difficulty, pay attention to timely guidance). 

2.5 Methods, Means and Matters Needing Attention 

Teaching Methods: Heuristic teaching, Example teaching, Analogy Teaching 
Teaching Means: The combination of classroom teaching and the use of multimedia courseware. Matters 
Needing Attention: The similarities and differences of condition and conclusion between Rolle's theorem and 
Lagrange theorem and the relationship between them; Rolle's theorem is a special case of the Lagrangian 
theorem; Lagrange theorem is generalization of Rolle's Theorem; 
note the mean value point of Rolle theorem and Lagrange mean value theorem is a point in the open interval, 
rather than an arbitrary point within the interval or a designated point, in other words, the two mean value 
theorem only "qualitatively" pointed out the existence of the median point, rather than "quantitatively” indicated 
the specific value of the median point; 
By learning the application of Rolle and Lagrange mean value theorem in this section as well as the 
application of each chapter in the future, repeatedly experience the significance and function of these 
theorems in calculus (Pan, 2014). 

2.6. Process 

Teaching content and teaching strategy: 
Review: 1) Rolle theorem If the function f(x) is satisfied: (1) be continuous on the closed interval [a,b], (2) be 
guided in the open interval (a,b),(3) be equal to the value of the function at the point of the interval, that is 
f(a)=f(b).Then there is at least one point (ab), in the interior (a,b), so that the derivative of the function f(x) 
at that point is equal to zero, that is f()=0 

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2) Geometric meaning: There is the tangent no perpendicular to the x axis at every point on the closed interval 
[a,b], and the connection of the two ends of the line and parallel to the x axis of the curve f(x), there is at least 
a point C, making its tangent parallel to the x axis. 

 

 
 

Figure 2: Geometric sketch of Rolle theorem 

Lagrange mean value theorem: (1) in practical application, the condition (3) of Rolle's Theorem cannot be 
satisfied, so its application is limited. If the condition (3) is removed, it is the Lagrange mean value theorem to 
be introduced. 
Lagrange mean value theorem: If the function f(x) is satisfied: (1) continuous on the closed interval [a,b], (2) 
be guided in the open interval (a,b), Then there is at least one point (a<<b), in the interior(a,b), so that f(b)-
f(a)=f’()(b-a). Before proving the Lagrange mean value theorem, the condition and the conclusion of the 
theorem are first known from the geometric point of view. 
Geometric meaning: The above (1) type can be deformed into 

ab

afbf
f






)()(
)(                                                                              (5) 

The slope of the right end of the equation is for the string AB, So there is no discontinuity in the interval [a,b] 
and each of its upper point is not perpendicular to the x axis of the tangent line, there is at least one point C, 
Making the tangent of the C point parallel to the string AB. When f(a)=f(b), the Lagrange theorem becomes 
Rolle theorem, Rolle theorem is a special case of the Lagrange mean value theorem and Lagrange mean 
value theorem is generalization of Rolle's Theorem, the following part is to prove Lagrange mean value 
theorem with Rolle theorem. 
The following part is trying to prove the Lagrange mean value theorem by application of Rolle theorem, which 
is inspired by the special relationship between Lagrange mean value theorem and Rolle theorem. 
Analysis: To prove that 
f(b)-f(a)=f’()(b-a) form, Is to permit the establishment of f’()=f(b)-f(a)(b-a),That is to prove the establishment 
of 

0
)()(

)( 





ab

afbf
f  .                                                                          (6) 

If the left of this equation is a function, For example, F (x) derivatives in point words, and then it becomes 
F()=0. And this form is the same to the form of conclusion of the theorem of Rolle. Therefore, it is necessary 
to find such a function f(x), if it can meet the conditions of Rolle's Theorem, we can prove Lagrange mean 
value theorem by using Rolle's Theorem. Then what kind of function is f(x). At this time, guide the students to 
observe the left of f()-((f(b)-f(a))/(b-a))=0, from the first item f(), naturally we thought of function f（x）, and 
by second item 

ab

afbf



 )()(
                                                                                               (7) 

to guide students to think of function 

x
ab

afbf



 )()(
,                                                                                   (8) 

328



In this way, the auxiliary function F(x)=f(x)-((f(b)-f(a))/(b-a))•x can be constructed. For function F(x) to first 
determine whether to meet the conditions of Rolle theorem, after verification, Function F (x) in the interval [a,b] 
does meet the conditions of Rolle theorem, then we can apply the Rolle theorem to prove the Lagrange mean 
value theorem. 
Prove: Constructing auxiliary function F(x)=f(x)-((f(b)-f(a))/(b-a))•x. We know that F(x) is continuous on the 
closed interval [a,b], F(x) can be guided in the open range (a,b), and F(a)=(bf(a)-af(b)/(b-a)=F(b). By Rolle's 
theorem, There is at least one point in the inside (a,b), so that F()=0, That is f()-((f(b)-f(a))/(b-a))=0, so then 
f(b)-f(a)= f’()(b-a). 
Theorem certificate. 

 

Figure 3: Comparison of geometric relations of differential mean value theorem 

Notes: 1) Constructing the auxiliary function method is a very important method to prove the mathematical 
proposition. The above is a method different from the teaching material proving the Lagrange mean value 
theorem by the constructing function. In the process, we first analyzed the conclusion and formed the auxiliary 
function F(x)=f(x)-((f(b)-f(a))/(b-a))•x. By using the Rolle theorem, Lagrange mean value theorem is proved. 
2) For at least one point in the theorem, the (1) type was established, which indicates that the mean value 
point must exist, but there may only be one, and there may be more than one, that is, the existence of the 

value point  may not be the only, and the value  is not specific. 
3) In the formula (1), if f(a)= f(b)，then f’()=0. Then the Lagrange mean value theorem becomes Rolle 
theorem, so here is the further explanation: Rolle's Theorem is a special case of the Lagrange mean value 
theorem and Lagrange mean value theorem is generalization of Rolle's Theorem 
Application of Lagrange mean value theorem: 1) can be used to prove the equation; 2) can be used to prove 
inequality. 
Example 1 Prove that if the derivative of the function f(x) in the interval I constants zero, then f(x) is a constant 
in the range I. 
Proof: Take any x1, x2I，might as well set x1x2，By condition f(x) is continuous on [x1,x2]，and is capable of 
guiding in (x1,x2)，then there is at least one point (ab) make the equation 

  
1221

)()( xxfxfxf    .                                                                  (9) 

Establish. from f(x)=0, then f()=0, f(x2)= f(x1) is from the upper. From the arbitrary nature of x1, x2, f(x) is a 
constant On I. 
The above is the application of Lagrange mean value theorem in equality proof, in fact, Lagrange mean value 
theorem can also be used to prove inequality. How to use the Lagrange mean value theorem to prove it? (Ask 
questions) to guide the students to think. 
Example 2 Prove when x0, (x/(1+x))Ln(1+x)x. 
Thinking and Deepening and Broadening the Teaching Content in this Section. Thinking questions：If the two 
conditions in the theorem are not satisfied, is the conclusion of the theorem still established? 
 
 
 



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3. Conclusion  

This paper, through research on the differential mean value theorem of teaching content, optimizing teaching 
design, application in the classroom teaching, gives the classroom teaching practice in the specific practices, 
thereby improving the effect of classroom teaching. The followings is reflection problem: whether the students 
have grasped the key points? As for the use of the multimedia courseware, whether the students recognize its 
assistance and positive effect? Whether the students accept the teaching strategies and the content of the 
teaching? Do the students have any other ways to construct the auxiliary function? 

Acknowledgments 

In the process of writing this article, I have got lots of support from my teachers and colleagues. They helped 
me to develop research ideas, modify the structure, content and format of the article, and they gave me 
inspiration and warm encouragement, for they put forward good opinions and suggestions. I feel quite grateful 
to them for their timely help. In addition, the funded projects related in this paper are as follows: Science and 
technology plan projects of Henan Province (152400410152); Key research projects in Henan province 
colleges and universities (15B120005). 

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