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Thread: mean value theorem

  1. #1
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    mean value theorem

    Use mean value theorem to prove :
    $\displaystyle \mid tan \ x + tan \ y \mid \geq \mid x+y \mid \ , \forall x,y \in (\frac {-\pi}{2}, \frac {\pi}{2}) $
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by flower3 View Post
    Use mean value theorem to prove :
    $\displaystyle \mid tan \ x + tan \ y \mid \geq \mid x+y \mid \ , \forall x,y \in (\frac {-\pi}{2}, \frac {\pi}{2}) $
    Suppose withou loss of generality that $\displaystyle u,v \in (-\pi/2,\pi.2)$ and $\displaystyle v>u$ then the MVT tells you that there is a $\displaystyle c$ in $\displaystyle (u,v)$ such that:

    $\displaystyle \left. \frac{d}{d\theta}\tan(\theta)\right|_{\theta=c}=\f rac{\tan(u)-\tan(v)}{u-v}$

    But

    $\displaystyle \frac{d}{d\theta}\tan(\theta) \ge 1 \ \ \ \theta \in (-\pi/2,\pi/2)$

    and you should using the fact that $\displaystyle \tan$ is odd be able to finish from there.

    CB
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