Mean Value Theorem proofs (1 Viewer)

leehuan

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So I got part a) out. Need only a small starter on part b). But it's mainly cause of the process.

Q: Use the MVT to prove that
a) ln(1+x)<x for x>0
b) -ln(1-x)<x/(1-x) for 0<x<1

So for part a) I defined f:[0,x], f(t)=ln(1+t) and used the MVT to get ln(1+x)/x=1/c (for some c in (0,x))
So cause c is positive ln(1+x)/x < 1 and x is positive so ln(1+x)<x. Nice and easy

But I'm only really concerned with defining the function. For x>0 I could just use [0,x].
How would I approach 0< x
< 1
?

Edit: This forum can't handle too many <'s it seems. Sorry
 

InteGrand

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So I got part a) out. Need only a small starter on part b). But it's mainly cause of the process.

Q: Use the MVT to prove that
a) ln(1+x)<x for x>0
b) -ln(1-x)<x/(1-x) for 0<x<1

So for part a) I defined f:[0,x], f(t)=ln(1+t) and used the MVT to get ln(1+x)/x=1/c (for some c in (0,x))
So cause c is positive ln(1+x)/x < 1 and x is positive so ln(1+x)<x. Nice and easy

But I'm only really concerned with defining the function. For x>0 I could just use [0,x].
How would I approach 0< x
< 1
?
For b), try defining f(t) = -ln(1-t) for 0 <= t<= x, where x is some arbitrary fixed number in (0,1).
 

kawaiipotato

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Can also be done by defining f(t) = (x-1)ln(1-x) for t E (0,1)
 

leehuan

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Oh yea I also used that method for some of the later questions lol
 

laters

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Can anyone prove using the mean value theorem? I've been trying to come up with functions that I can apply the MVT to but I've gotten stuck.

Also for
 

leehuan

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Can anyone prove using the mean value theorem? I've been trying to come up with functions that I can apply the MVT to but I've gotten stuck.

Also for
Basically what InteGrand said, but it actually takes nothing more than a tiny bit of intuition to figure out part d). It's just arctan over [1, 9/8] (Or you can do arctan [1,x] and then let x=9/8 but there's no need)
 

laters

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Thanks leehuan and Integrand!! I managed to get the arctan one out.

However for the first one I asked, I just want some clarification. I know 23 < c < 27 so if c < 27 then I can obtain the lower bound. But obviously if I use c>23 then I would require a calculator to show that the 3-23^(1/3)< 0.16..< 3/16, which defeats the purpose of the question. c>512/27 would allow this, but I didn't apply the MVT in (512/27, 27). What did you do?
 
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InteGrand

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Thanks leehuan and Integrand!! I managed to get the arctan one out.

However for the first one I asked, I just want some clarification. I know 23<c<27 so if c<27 so I can obtain the lower bound. But obviously if I use c>23 then I would require a calculator to show that the 3-23^(1/3)<0.16..<3/16, which defeats the purpose of the question. c>512/27 would allow this, but I didn't apply the MVT in (512/27, 27). What did you do?
We don't need a calculator.





 
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leehuan

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Please start me off on the processes in the inductive step.



I have no clue how to apply the hypothesis here...
 

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