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Monotonically increasing function

  1. Nov 22, 2013 #1
    I have the following function

    [itex]f=\frac{B}{y^{3}}+\frac{C}{y^{4}}\mid\frac{dy}{dx}\mid[/itex]

    where [itex]B[/itex] and [itex]C[/itex] are constants and where [itex]y[/itex] is a monotonically
    decreasing function of [itex]x[/itex] ([itex]\mid\frac{dy}{dx}\mid[/itex] stands for absolute value of derivative). According to my model, all
    signs indicate that [itex]f[/itex] is a monotonically increasing function of
    [itex]x[/itex]. Numerical experiments and logical arguments confirm this but
    I need a rigorous proof of this. If [itex]f[/itex] is not unconditionally monotonically
    increasing function of [itex]x[/itex] I wish to know under what conditions it
    will be monotonically increasing function of [itex]x[/itex].
     
  2. jcsd
  3. Nov 22, 2013 #2

    pasmith

    User Avatar
    Homework Helper

    If [itex]y' \leq 0[/itex] then [itex]|y'| = -y'[/itex], so you have
    [tex]
    f = By^{-3} - Cy^{-4}y'.
    [/tex]
    Now differentiate, and find conditions necessary for [itex]f' \geq 0[/itex].
     
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