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RolyPoly

  • 3 years ago

How to get started? Consider \(\ln x \) in [1, 1+t] (t>0), show that \( \frac{x}{x+1} < \ln (1+x) < x\) .

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  1. mukushla
    • 3 years ago
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    start with defining a function for each part for example\[f(x)=\ln(1+x)-\frac{x}{x+1}\]and show that f(x) is strictly increasing (for the other part decreasing)

  2. mukushla
    • 3 years ago
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    make sense?

  3. RolyPoly
    • 3 years ago
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    I'm sorry. I don't quite understand. Why are you defining that function? Or, I should ask what you are doing. (Sorry again!)

  4. mukushla
    • 3 years ago
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    see u have 2 works to do here one is showing\[\ln(1+x)>\frac{x}{1+x}\]or\[\ln(1+x)-\frac{x}{1+x}>0\]so defining\[f(x)=\ln(1+x)-\frac{x}{x+1}\]and showing that f(x) is increasing is what u need here

  5. mukushla
    • 3 years ago
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    other part is showing\[\ln(1+x)<x\]or\[x-\ln(1+x)>0\]for this one let\[g(x)=x-\ln(1+x)\]and show that g(x) is strictly increasing

  6. RolyPoly
    • 3 years ago
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    (Shhh~~~) Would you mind letting me try the second half without giving me the hint? Thanks for your help. I start to understand it!

  7. mukushla
    • 3 years ago
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    yeah im finished here :)

  8. RolyPoly
    • 3 years ago
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    But wait. If I'm doing this way, why do I have to consider lnx in the interval [1, 1+t] for t>0?

  9. RolyPoly
    • 3 years ago
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    @mukushla Sorry! I'm calling you back :P

  10. mukushla
    • 3 years ago
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    no problem rolypoly :) yeah thats the domain for which we want prove that inequality

  11. RolyPoly
    • 3 years ago
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    Two questions: 1. To show f(x) is increasing, we do differentiation, right? (First derivative) 2. If we just take derivative, we're not really consider the domain, and also lnx.

  12. mukushla
    • 3 years ago
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    1. right and showing f'(x)>0 2. we should just consider the given interval [1,1+t]

  13. RolyPoly
    • 3 years ago
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    How can we ''just consider the given interval''?

  14. mukushla
    • 3 years ago
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    emm...sorry what do u mean?

  15. RolyPoly
    • 3 years ago
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    Sorry! You said that we should just consider the given interval. But in the calculation, how can we (show that) we just consider the given interval? Since we're doing it in a ''general'' way.

  16. mukushla
    • 3 years ago
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    ahh yeah .. when u want to show that f'(x)>0 u must use that

  17. RolyPoly
    • 3 years ago
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    \[f'(x) = \frac{1}{1+x} - \frac{(x+1) -(x)}{(x+1)^2}=\frac{1}{1+x}-\frac{1}{(x+1)^2} = \frac{x}{(x+1)^2}\] Hmm, so far so good?

  18. mukushla
    • 3 years ago
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    yes :)

  19. RolyPoly
    • 3 years ago
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    Then, it's bad. Once I put f'(x) = 0, I get x=0

  20. mukushla
    • 3 years ago
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    u see f'(x)>0 for given interval

  21. mukushla
    • 3 years ago
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    why letting it equal to zero?

  22. RolyPoly
    • 3 years ago
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    Errr, usual practice.

  23. RolyPoly
    • 3 years ago
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    Oh wait. For the given interval, f'(x)>0 since 1 is positive and 1+t is also positive for t>0. So, f'(x) >0

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