Carissa15
  • Carissa15
If anyone could me, I don't know what the mean value theorem is but I have a question about it as below.
Mathematics
katieb
  • katieb
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Carissa15
  • Carissa15
Apply the mean value Theorem to f(x)=In(1+x) to show that \[\frac{ x }{ 1+x }<\ln (1+x)0.\]
freckles
  • freckles
\[\text{ try something like this } \\ \text{ use the mvt for } f(x)=\ln(1+x) \text{ on } (0,x)\]
Carissa15
  • Carissa15
I am not sure how to use the mean value theorem

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freckles
  • freckles
the function f(x)=ln(x+1) is continuous on [0,1] and differentiable on (0,1) so there exist \[c \in (0,x) \text{ such that } f'(c)=\frac{f(x)-f(0)}{x-0}\]
freckles
  • freckles
and you are definitely going make use that c is in the interval (0,x) that is 0
freckles
  • freckles
the function f(x)=ln(x+1) is continuous on [0,x] and differentiable on (0,x)*
freckles
  • freckles
anyways let me know if you still need help
Carissa15
  • Carissa15
Thank you, so all it is asking is to prove that the function will be continuous and differentiable from the values using the mvt?
freckles
  • freckles
no you have to show what it asked
freckles
  • freckles
use the thing above find f'(c) and...f(x) and f(0)
freckles
  • freckles
\[f(x)=\ln(x+1)\\ \text{ can you find } f'(x)?\]
freckles
  • freckles
not sure if you are there or not but I have to go
DanJS
  • DanJS
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
DanJS
  • DanJS
It is an existence type thing
DanJS
  • DanJS
|dw:1438761461750:dw|
DanJS
  • DanJS
something like that..
Carissa15
  • Carissa15
thank you, i will have to come back to this. thanks
Carissa15
  • Carissa15
Thank you all for your help
Carissa15
  • Carissa15
I am still unsure of this, I think I need to substitute any x>0 for each of the 3 equations \[\frac{ x }{ 1+x }<\ln(1+x) 0 each equation will be less than the last?
freckles
  • freckles
we can use that formula in the mvt since our function satisfies that f is continuous on [0,x] and differentiable on (0,x) I posted above this means we have some number c in (0,x) such that: \[f'(c)=\frac{f(x)-f(0)}{x-0} \\ \text{ where } f(x)=\ln(x+1)\] You need to find f'(x)... Then plug in c into that so you can have your left hand side. Then evaluate f(0) and replace f(0) with whatever you get for that. Then use the inequality 0c.
Carissa15
  • Carissa15
Great, thank you so much. :-)

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