I am faced with the following question:
Use the mean value theorem to establish bounds in the following cases.
(a) For $-\ln(1-y)$, by considering $\ln x$ in the range $0<1-y<x<1$.
(b) For $e^y-1$, by considering $e^x-1$ in the range $0<x<y$.
I understand the definition of the Mean Value Theorem — my problem is I do not understand what the question is asking!
What are we trying to "establish bounds" for? For the values of the functions (e.g. $-\ln(1-y)$)? Or for the values of the derivatives? And what do they mean by "considering [another function]"? And isn't e.g. $0<x<y$ the domain of the functions, not their range?
Maybe I am missing something obvious. You can provide solutions if you wish, but mostly I am looking for insights into understanding the question.
(The question is exercise 2.26 in Mathematical Methods for Physics and Engineering, by Riley, Hobson, and Bence.)