# Thread: mean value theorem as an obvious result ?

1. ## mean value theorem as an obvious result ?

mean value theorem
This theorem can be understood intuitively by applying it to motion: If a car travels one hundred miles in one hour, then its average speed during that time was 100 miles per hour. To get at that average speed, the car either has to go at a constant 100 miles per hour during that whole time, or, if it goes slower at one moment, it has to go faster at another moment as well (and vice versa), in order to still end up with an average of 100 miles per hour. Therefore, the Mean Value Theorem tells us that at some point during the journey, the car must have been traveling at exactly 100 miles per hour; that is, it was traveling at its average speed.

sir my doubt is that this theorem seems to be an obvious result in fact if our car stops at 120 km/h and its average velocity is 100 miles per hour then speedometer will read each and every reading up to 120. therefore speedometer will also read 100 .then what is special about mean value theorem,
what is the need of proving it .is there is any practical example where we need the help of this proof.

2. Under your conditions, it's not necessary that the car make it all the way to 120 km/h. It depends on how long the car is going at slower speeds. If the car speeds up very quickly to a little over 100, in order to make up for the slow speeds, then theoretically, you wouldn't need to go as high as 120 km/h.

The real utility of the mean value theorem is in proving the Fundamental Theorem of the Calculus, which is by far the most important theorem in all of mathematics: it is responsible for the modern technological age.