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.
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).
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