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.

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.
