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Math Help - Prove that

  1. #1
    Super Member dhiab's Avatar
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    Prove that

    Prove that :  \forall x > 0:1 - \frac{{x^2 }}{2} \le \cos x \le 1<br /> <br />
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  2. #2
    MHF Contributor red_dog's Avatar
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    We have to prove the first inequality. The second is obviously.

    Let f(x)=1-\frac{x^2}{2}-\cos x, \ x>0.

    f'(x)=-x+\sin x

    f''(x)=-1+\cos x\leq 0, \forall x\in\mathbb{R}\Rightarrow f' is decreasing.

    Then, \forall x>0\Rightarrow f'(x)\leq f'(0)=0\Rightarrow f is decreasing.

    Then, \forall x>0\Rightarrow f(x)<\leq f(0)=0\Rightarrow 1-\frac{x^2}{2}\leq \cos x
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  3. #3
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    Quote Originally Posted by dhiab View Post
    Prove that :  \forall x > 0:1 - \frac{{x^2 }}{2} \le \cos x \le 1<br /> <br />
    This is equivalent to proving

     |x|\geq |\sin x | \geq 0
    Last edited by mr fantastic; September 18th 2009 at 09:00 AM. Reason: Restored original reply
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