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Math Help - mean value theorem

  1. #1
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    mean value theorem

    Use mean value theorem to prove :
     \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 :
     \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 u,v \in (-\pi/2,\pi.2) and v>u then the MVT tells you that there is a c in (u,v) such that:

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

    But

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

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

    CB
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