Importance of Rolle's and Lagrange's theorem in daily life: for school children

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.

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.

You can explain Rolle's theorem by saying that if your average speed during a journey from A to B was say 50kms/hour then there had to be a time when your instantaneous speed was 50kms/hour as well. Significance of LMVT.