Question

what is means value theorem what is its propase hoe we can use in real life

Answer 1

There's a fairly good explanation of the MVT here:

http://www.sosmath.com/calculus/diff/der11/der11.html

Answer 2

Leibniz and L'Hospital rules use it, Taylor series is proven using it

Answer 3

in real life it says if you average 50 mph on your journey at some point you must have been going exactly 50 mph

Answer 4

Very nice and short explanation @satellite73....I like it.

Answer 5

mean value theorem gives surety that a critical point is always there between the two points of function whose values are equal (special case of mean value theorem known as Rolle's theorem)
it is actually kind of defination of slope that if you have two points (a,f(a)) b(,f(b)) then if you draw a secant line connecting a and b if there is third point btween a and called c and you draw a tangent line at c this theorem gurantee that both lines must be parallel