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Math Help - Mean value theorem

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    Mean value theorem

    By the Mean value theorem show that:

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

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

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

    Then by differentiating both sides

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

    1 > 1 + x^2

    x^2 < 0.


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

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