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Thread: More Proofs

  1. #1
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    More Proofs

    Let $\displaystyle \alpha > 0$. Prove that no matter how small $\displaystyle \alpha$ is, there is an $\displaystyle N \epsilon$ R such that $\displaystyle ln x \leq x^{\alpha}$ for all $\displaystyle x \leq N$. Note: N depends on $\displaystyle \alpha$.

    I need to use the mean value theorem to do this.
    I started by finding the difference of the derivatives of both functions.
    Then didnt know where to go... Any help would be greatly appreciated...
    Last edited by Caity; Nov 21st 2008 at 08:55 AM.
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  2. #2
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    Fixed Latex error... sorry about that
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  3. #3
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    I've been playing around with this problem... This is what I have so far...

    $\displaystyle g' (x) - f' (x)$ is decreasing when $\displaystyle \alpha $ is very small.

    For $\displaystyle \alpha = \frac{1}{200}$ or smaller $\displaystyle x = 2.27$

    The inequality only holds true for $\displaystyle x\leq 2.27 $and I need to find a $\displaystyle N\leq x$

    Can anyone help me with explaining this or let me know if I am doing something wrong please??
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