The discussion revolves around understanding the Mean Value Theorem and its relationship to Rolle's Theorem, as well as addressing specific statements related to these theorems. Participants are examining the correctness of various assertions made in a homework problem.
Some participants have provided insights into the definitions and conditions of the theorems, while others express uncertainty about specific statements. There is an ongoing exploration of the nuances in the definitions, particularly regarding continuity at the endpoints of the interval.
There is mention of textbook references and a library item that may clarify the definitions, indicating that participants are working within the constraints of their course materials and homework guidelines.
VeeEight said:(1) is Rolle's Theorem
(2) is the Mean Value Theorem, which is a generalization of Rolle's Theorem
I did not know (3) but that is interesting
(4) is correct, f(x) can be a constant function not equal to 0.
(5) is correct - take a(x) = f(x) - g(x). It's derivative is 0, so a(x) is constant (you can prove this using MVT).
VeeEight said:Oh sorry, I missed that
(1) states that f is defined on [a,b] but it is not necessarily continuous there. It is continuous on (a,b) (since it is differentiable there) but not necessarily at a or b.