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

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. 
