Question

If anyone could me, I don't know what the mean value theorem is but I have a question about it as below.

Answer 1

Apply the mean value Theorem to f(x)=In(1+x) to show that \[\frac{ x }{ 1+x }<\ln (1+x)<x, for x>0.\]

Answer 2

\[\text{ try something like this } \\ \text{ use the mvt for } f(x)=\ln(1+x) \text{ on } (0,x)\]

Answer 3

I am not sure how to use the mean value theorem

Answer 4

the function f(x)=ln(x+1) is continuous on [0,1] and differentiable on (0,1)
so there exist
\[c \in (0,x) \text{ such that } f'(c)=\frac{f(x)-f(0)}{x-0}\]

Answer 5

and you are definitely going make use that c is in the interval (0,x)
that is 0<c<x <--this will be handy inequality in your proof

Answer 6

the function f(x)=ln(x+1) is continuous on [0,x] and differentiable on (0,x)*

Answer 7

anyways let me know if you still need help

Answer 8

Thank you, so all it is asking is to prove that the function will be continuous and differentiable from the values using the mvt?

Answer 9

no you have to show what it asked

Answer 10

use the thing above
find f'(c)
and...f(x) and f(0)

Answer 11

\[f(x)=\ln(x+1)\\ \text{ can you find } f'(x)?\]

Answer 12

not sure if you are there or not
but I have to go

Answer 13

The mean value theorem basically says, if you have a continuous function between two x values,
There is some point in between the interval ends where the slope of the tangent line to the function is the same as the slope of a secant line connecting the endpoints...if i remember right

Answer 14

It is an existence type thing