In many textbooks, the former two have been proved with the help of Rolle's theorem. However my teacher(and many sites as well) say that Rolle's theorem is a special case of LMVT and Cauchy's is a generalization.
Can we prove Cauchy's MVT and LMVT without using Rolle's theorem? If not, should we admit that these are just applications of Rolle's theorem and hence yield no extra result?
I found an analogy which could be useful:
Suppose a car is travelling at an average speed of 40 miles/hr. In the course of the travel, it has to, at some point, travel at exactly 40 miles an hour.
This is exactly what LMVT has to say.