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Thread: Monotonically increasing function

  1. #1
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    Monotonically increasing function

    I have the following function

    $\displaystyle f=\frac{B}{y^{3}}+\frac{C}{y^{4}}\mid\frac{dy}{dx} \mid$

    where $\displaystyle B$ and $\displaystyle C$ are constants and where $\displaystyle y$ is a monotonically
    decreasing function of $\displaystyle x$ ($\displaystyle \mid\frac{dy}{dx}\mid$ stands for absolute value of derivative). According to my model, all
    signs indicate that $\displaystyle f$ is a monotonically increasing function of
    $\displaystyle x$. Numerical experiments and logical arguments confirm this but
    I need a rigorous proof of this. If $\displaystyle f$ is not unconditionally monotonically
    increasing function of $\displaystyle x$ I wish to know under what conditions it
    will be monotonically increasing function of $\displaystyle x$.
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  2. #2
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    Re: Monotonically increasing function

    y=cosx is monotonically decreasing from 0 to 180.
    y=-sinx
    IyI=sinx which is increasing from 0 to 90 and decreasing from 90 to 180.

    dIyI/dx needs to be positive. For above example,
    dIyI/dx=cosx which is positive from 0 to 90.

    a,b positive: b<a -> b3<a3 & b4<a4
    a,b negative: b<a -> b3<a3 & b4>a4
    Last edited by Hartlw; Nov 22nd 2013 at 04:02 AM.
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