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Math Help - solving inequality.

  1. #1
    Vim
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    solving inequality.

    hi everyone. first of all i have to apologize for my poor English.
    I've got an inequality which I know is true but don't know how to solve,can anyone help?
    2^{a}-2^{-a} \neq 2^{b}-2^{-b}
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  2. #2
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    Re: solving inequality.

    That isn't an inequality.

    Anyway, 2^a-2^{-a} = 2^b-2^{-b} \Leftrightarrow a=b.

    I.

    Let's consider the functions f:\mathbb{R} \rightarrow \mathbb{R}, f(x)=2^x-2^{-x}, f_1, f_2:\mathbb{R}\rightarrow \mathbb{R}, f_1(x)=2^x, f_2(x)=2^{-x}=\left ( \frac{1}{2} \right )^x.
    f_1 is strictly increasing;
    f_2 is strictly decreasing, so -f_2 is strictly increasing;

    f=f_1+f_2 is strictly increasing (being a sum of two strictly increasing functions) \Rightarrow f is injective, and therefore f(a)=f(b) \Leftrightarrow a=b.


    II.

    f:\mathbb{R} \rightarrow \mathbb{R}, f(x)=2^x-2^{-x}

    f'(x)=(2^x-2^{-x})'=2^x\ln 2+2^{-x}\ln 2=\ln2(2^x+2^{-x})> 0, for any x\in \mathbb{R} \Rightarrow f is injective, and therefore f(a)=f(b) \Leftrightarrow a=b.
    Last edited by veileen; November 17th 2012 at 06:05 AM.
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