Understanding Mean Value Theorem: Solving Homework Problems

[Slimsta]

Homework Statement

http://img14.imageshack.us/img14/6132/proiqc.jpg

Homework Equations

The Attempt at a Solution

the first 3 are from the textbook so they must be right.. the last 2 I am pretty sure i got right too..
because the 4th one, if f'(x)=0 then f(x)= c .. so its false.
im not too sure about the 5th one but I am 90% sure it should true.

 

[VeeEight]

(1) is Rolle's Theorem (Edit: this is incorrect - see the following posts)
(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).

 

[Slimsta]

(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).

i know that 1 and 2 are Rolle's Theorem and Mean Value Theorem, and its written exactly like in my textbook.. i can't find the mistake.

https://www.physicsforums.com/library.php?do=view_item&itemid=231 its even stated here..

so what is wrong?

 

[VeeEight]

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.

 

[Slimsta]

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.

thats a tricky question.. weird.
thanks a lot for your help man!

 

[VeeEight]

No problem, can't believe I missed that
In my experience, the physical science courses are where they try to 'trick' you like that, I've never had any questions like that when I was a math undergrad. I guess now you've learned that you have to pay attention to every little detail when writing tests in this class (which is probably a good idea, regardless). Cheers.