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Thread: Mean value theorem

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    Mean value theorem

    By the Mean value theorem show that:

    $\displaystyle \arctan{x} \leq x\ for\ x \geq 0$
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  2. #2
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    Quote Originally Posted by acevipa View Post
    By the Mean value theorem show that:

    $\displaystyle \arctan{x} \leq x\ for\ x \geq 0$
    Assume that

    $\displaystyle \arctan{x} > x$ for $\displaystyle x \geq 0$.

    Then by differentiating both sides

    $\displaystyle \frac{1}{1 + x^2} > 1$

    $\displaystyle 1 > 1 + x^2$

    $\displaystyle x^2 < 0$.


    But $\displaystyle x^2 \geq 0$ for all $\displaystyle x \in \mathbf{R}$. So we have arrived at a contradiction. So our original statement that $\displaystyle \arctan{x} > x$ is wrong.

    Therefore $\displaystyle \arctan{x} \leq x$ for all $\displaystyle x \geq 0$.
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