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Math Help - inequality problem

  1. #1
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    Unhappy inequality problem

    Hi all,

    I can't find the answer for this question...

    Find the region of x such that arcsin (x ) < x + (x^3)/6 holds.

    I try differentiate x + (x^3)/6 - arcsin (x) and want to find the roots but it didn't work...


    Please help.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by dimeola View Post
    Hi all,

    I can't find the answer for this question...

    Find the region of x such that arcsin (x ) < x + (x^3)/6 holds.

    I try differentiate x + (x^3)/6 - arcsin (x) and want to find the roots but it didn't work...


    Please help.
    put x=\sin t and f(t)=t - \sin t - \frac{\sin^3 t}{6}, \ \ \frac{-\pi}{2} \leq t \leq \frac{\pi}{2}. see that f(0)=0, \ f(-\pi/2) < 0, \ f(\pi/2) > 0. we're looking for values of t for which f(t) < 0.

    we have: f'(t)=1 - \cos t - \frac{\cos t \sin ^2 t}{2}=\frac{(1- \cos t)^2(2 + \cos t)}{2} \geq 0. you should be able to see now that -1 \leq x < 0 is the answer.
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  3. #3
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    Hi,

    Thank you very much
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