Verifying Hypotheses of the Mean Value Theorem for f(x)=1/(x-2) on [1,4]

[FritoTaco]

Homework Statement

Find all the numbers c that satisfy the conclusion of the Mean Value Theorem for the functions

f(x)=\dfrac{1}{x-2} on the interval [1, 4]

f(x)=\dfrac{1}{x-2} on the interval [3, 6]

I don't need help solving for c, I just want to know how I can verify that the hypotheses of the mean Value Theorem are satisfied by the function f(x) on the given interval. I know the first one, there is no such number c that is guaranteed by the mean Value Theorem but there is for the second one. How can we verify the first one?

Homework Equations

Rolle's Theorem: Let f be a function that satisfies the following three hypotheses:

1. f is continuous on the closed interval [a, b]
2. f is differentiable on the open interval (a, b)
3. f(a)=f(b)

Then there exists a number c between a and b such that f'(c)=0

The Attempt at a Solution

So I would test if it's differentiable for the first function.

f(1)=\dfrac{1}{1-2}=-1

f(4)=\dfrac{1}{4-2}=1/2

So for the first interval of [1, 4] there is no such c number because the inputs (-1 and 1/2) are not equal.
How do we know if it's continuous? If it's differentiable does that mean it has to be continuous?

 

[mfb]

The mean value theorem is more general than Rolle's theorem, it does not need the f(a)=f(b) condition.

If it's differentiable does that mean it has to be continuous?

Yes - but keep in mind that this just helps directly with (a,b), not with [a,b].

 

[FritoTaco]

So my interval is a closed interval for [a, b]. If the case doesn't apply, how would you go about verifying if it's continuous?

 

[mfb]

At the time you get those questions, it should be fine to do that just by inspection.

But there is also a statement about the continuity of f(x)=c/g(x) with a constant c that depends on properties of g(x).

 

[FritoTaco]

Alright, thank you!