I am school teacher and teaching math. I know the Rolle's theorem and Lagrange's theorem. I can solve the problems numerically then and there. However, I was completely failed to explain the significance or applications of these theorems by practical.
If this site can help me to explain both of the theorems with good practical examples are enough and I can teach my students well. Awaiting your replies.
Since Rolle's theorem asserts the existence of a point where the derivative vanishes, I assume your students already know basic notions like continuity and differentiability. One way to illustrate the theorem in terms of a practical example is to look at the calendar listing the precise time for sunset each day. One notices that around the precise date in the summer when sunset is the latest, the precise hour changes very little from day to day in the vicinity of the precise date. This is an illustration of Rolle's theorem because near a point where the derivative vanishes, the function changes very little.
Here's a contrarian answer. I don't think Rolle's or Lagrange's theorems have any immediate practical value and can't be justified that way. You should not try.
There is no doubt that students should learn about the practical value of calculus. Much of that value is that calculus is the language of change. The derivative provides a local linear approximation to a smooth function, so it's $0$ at an extreme point. You can make better local approximations with higher derivatives and higher order polynomials. You can construct genuinely interesting toy applications with these ideas and even touch on truly practical ones.
Rolle's theorem is intuitively obvious. From the Brittanica encyclopedia:
Other than being useful in proving the mean-value theorem, Rolle’s theorem is seldom used, since it establishes only the existence of a solution and not its value.
( @littleO hints at these ideas in a comment at the question.)
Maths is a topic that's useful for everything and nothing. I use it a lot on the job where I work. I work in an oil analysis laboratory (Its kind of like a pathology for trucks if you can imagine that.) We test oil samples to see if there is a problem with the truck. Our lab tests generate a lot of data and there's a lot of opportunities to apply mathematics and statistics to our work, and we do.
We also use robots in our lab to automate many of our tasks. We use image processing to recognize bar codes and spot bubbles in oil. There's a LOT of calculus and mathematics behind these tasks.
As for specific applications, these theorems are used in developing numerical methods of solving equations. See here to get the falvour.
I guess the bisection method would be an example of something similar that seems more practical.
Probably the most useful application of these theorems is to give students a better understanding of calculus which is used in so many different areas of our lives.
In the information age, more data is being collected on our lives and there are more and more opportunities to apply mathematics to understand that data.
EDIT
Kindly discuss still more with good examples with graphical presentation. So that they can understand more
I think its better to talk about applications of calculus. While there may be direct applications of the Rolle's and Lagrange's theorems, the main reason we study them is to better understand calculus.
At work I use calculus to spot bubbles of water in oil. See here for pretty pictures.
Image processing has lots of calculus and its used in many different fields such as medical imaging (CT scans, MRI, diagnosing sick patients), industrial automation (quality control, bubble detection etc ..) and security (I have a friend who worked in facial recognition software for airports).
For those interested in computer games. Calculus is used in physics engines which keep track of moving objects in game.
Its also used in computer graphics to give you very shiny realistic looking modern computer games and scientific visualization.
Calculus is used in neural networks and control theory which are used for many things including controlling industrial processes and more importantly ... teaching robots to play soccer :)
This could become a very long list but these are some of the areas that I have some personal experience in, hope they help.
There are plenty of questions concerning the theorem of Rolle or Lagrange. Interesting might be different formulations to You and the level of school teaching. For example this answer: other forms of the Lagrange theorem or this lagrange theorem demonstration.
Such general theorems offer many applications and formulations. They can be made more open or closed depending on the very intents of the lessons. Minima and maxima are as powerful and continuity and differentiability too. This is an open-ended question. Conditions are open to many variations and for the introduction of many other terms and functions and situations in math. For example how to prove with Lagrange theorem. Gives a lesson with the character of mathematical proof.
Or this one opens this at this time familiar theorems to set theory: lagrange theorem and cyclic subgroups.
This question for example targets the limits of Rolle theorem. This discusses which theorem came first Rolle or mean value the timely order in a lesson sequence and the dependence of both and what is needed in which situation. Here an example with a contraction on a polynomial of degree three: rolles theorem contradiction.
Statistics show 2,360 results for the Rolle theorem on math.stackexchange.com and Lagrange theorem has 589 results. So these are rather frequent in the questions and answers. And the amount is filling more than a book.
These are at most applications. A good site for definitions is Lagrange's theorem and Lagrange's theorem (group theory) or Mean value theorem the version for calculus. This is already a great source of inspiration.
This has the sections:
2 Formal statement 3 Proof 4 Implications 5 Cauchy's mean value theorem 6 Generalization for determinants 7 Mean value theorem in several variables 8 Mean value theorem for vector-valued functions 9 Mean value theorems for definite integrals 10 A probabilistic analog of the mean value theorem 11 Generalization in complex analysis
That is already enough for three months or more. They refine that only the formal statements are also known as Lagrange's Mean Value Theorem. They show proof not so super duper but an interesting redefinition of the theorem and refinements. They show up the difference between Cauchys and Lagrange's version is a generalized case.
They have nice references and extra pages concerning the mean value theorem with divided differences.
So this question is totally cruel broad and open and can not be even filled with rich or simple examples because of the many perspectives this has in different areas of modern Mathematics. For the school lessons, it might be a very important reduction to concentrate on the proof. But more important is that the wikipedia.org page includes the theorem of Rolle and has an extra page for that.
A problem that this has is modernness with sets and the group's theoretical formulation. These are the most interesting ones because the theorem holds all modern mathematics has to be related to the Zermelo - Frankel set theorems and the axioms of choice and beyond that be open to n-categories from the 0-category are sets.
Most modern and advanced teachers have the challenge to transfer school math to include this newest foundation since the target of the school of the future is to educate the present day's competencies with the furthest reaches.
Re Lagrange's theorem: This theorem was developed by people who used it to systematise their knowledge of lots of examples, and became part of the notion that group theory teaching did not need lots of examples. Better to study groups of symmetries, multiplication tables, and their look for subgroups, and normal subgroups, do some generators and relations, .... I've even given a talk on the Todd-Coxeter method for 14 year olds, presented as a game with words!
Rolle's theorem: this should be part of the notion of rate of change. A good everyday (or every other day!) example is the rate of change in sunset times over the seasons.
Rolles Theorem Example: When I throw a ball vertically up, its initial displacement zero ($f(a)=0$) and when I catch it again its displacement is zero ($f(b)=0$). As displacement function satisfy criteria of rolles theorem of continuity and differentiability over $(a, b)$ interval. There exists at least one $f'(c)= 0$, and in fact velocity is zero at the highest point. Here $f(x)$ is the height of ball and $f'(x)$ is the vertical velocity of the ball.
