1. ## Lagrange's Theorem

Hi, I have uploaded a picture of two proved prepositions. They used Lagrange, and I have trouble understanding why does using lagrange proves the condition for every point between x and y. Can someone explain that? Thank you.

2. ## Re: Lagrange's Theorem

You don't say what your difficulty is and it seems fairly clear to me. Perhaps it would make more sense to phrase it a little differently. Suppose f'(c)= 0 for all c in I. Then for any two numbers, a and b, in I, the mean value theorem says "there exist c in I such that $\displaystyle \frac{f(a)- f(b)}{a- b}= f'(c)= 0$. From that it follows that f(a)- f(b)= 0(a- b)= 0 so that f(a)= f(b). That is, for any two numbers a and b in I, f(a)=n f(b).

Suppose f'(c)> 0 for all c in I. For b>a, by the mean value theorem there exist c in I such that $\displaystyle \frac{f(b)- f(a)}{b- a}= f'(c)$ so that f(b)- f(a)= f'(c)(b-a) f'(c)> 0 by hypothesis and b- a> 0 because b> a. Therefore f'(c)(b- a)> 0 so f(b)- f(a)> 0 and f(b)> f(a).

3. ## Re: Lagrange's Theorem

Wait minute , so that a and b are any points in the interval? They are not the end points of the interval? If that's the case, I understand completely. Because when we learned lagrange they gave the interval [a,b] and did f(a)-f(b)/(a-b)=f^1(c) so that's why i am confused