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!