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Math Help - How to prove sinx+3-2x=0 has only one unique solution

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    How to prove sinx+3-2x=0 has only one unique solution

    I know sinx +3-2x=0 has only one unique solution that i found using Newtons algorithm in excel and from seeing the graph of sinx+3-2x=0 but i do not know how to prove this unique solution algebraically. Hopefully someone can help.

    Thankyou
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    MHF Contributor Also sprach Zarathustra's Avatar
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    First define the function f(x)=sinx - 2x+3, this is continuous and differentiable function for all x.

    Now, f(0)=sin0-2*0+3>0 and f(6)=sin(6)-12+3 ~ 1-12+3<0, from the Intermediate value theorem follows that there exist x_0 in (0,6) so that f(x_0)=0, hence we know that f(x) have one zero.


    Now we will prove that above zero is unique!

    We look at the derivative of f(x).

    f'(x)=cosx-2<0 for all x. In other words the function f(x) is decreasing for every x, therefor the function is one-to-one, hence f(x)=0 only at x_0.
    Last edited by Also sprach Zarathustra; November 21st 2010 at 12:11 PM.
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    Quote Originally Posted by nicolas123 View Post
    I know sinx +3-2x=0 has only one unique solution that i found using Newtons algorithm in excel and from seeing the graph of sinx+3-2x=0
    The derivative of \sin(x)+3-2x is \cos(x)-2 which is always negative.
    The fact we have a decreasing function tells us what?
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    Thanks so sinx-2x=3=0 has one solution at the point 1.916 and then to prove it has one unique solution is that the first derivative cos(x)-2 decreases for every increasing value of x.
    Cheers.
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    A decreasing function is one-to-one.
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    Quote Originally Posted by nicolas123 View Post
    I know sinx +3-2x=0 has only one unique solution that i found using Newtons algorithm in excel and from seeing the graph of sinx+3-2x=0 but i do not know how to prove this unique solution algebraically. Hopefully someone can help.

    Thankyou
    Alternatively,

    sinx=2x-3

    The derivative of sinx is cosx, whose maximum value is 1.

    The slope of the line 2x-3 is 2, hence the sinewave can never cross over it a second time (the line slope would need to be <1 for this to be possible).
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