- #1

FritoTaco

- 132

- 23

## Homework Statement

Find all the numbers [itex]c[/itex] that satisfy the conclusion of the Mean Value Theorem for the functions

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

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

I don't need help solving for [itex]c[/itex], I just want to know how I can verify that the hypotheses of the mean Value Theorem are satisfied by the function [itex]f(x)[/itex] on the given interval. I know the first one, there is no such number [itex]c[/itex] 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 [itex]f[/itex] be a function that satisfies the following three hypotheses:

1. [itex]f[/itex] is continuous on the closed interval [itex][a, b][/itex]

2. [itex]f[/itex] is differentiable on the open interval [itex](a, b)[/itex]

3. [itex]f(a)=f(b)[/itex]

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

## The Attempt at a Solution

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

[itex]f(1)=\dfrac{1}{1-2}=-1[/itex]

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

So for the first interval of [itex][1, 4][/itex] there is no such [itex]c[/itex] 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?