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Function help.

  1. Aug 30, 2010 #1

    Mentallic

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    1. The problem statement, all variables and given/known data
    I need to find n such that,

    [tex]y=\frac{ln(cos(x))}{cos(1)}-x^n[/tex]

    is zero for all [tex]0\leq x\leq 1[/tex].


    3. The attempt at a solution
    I've already narrowed it down to [tex]2<n<2.5[/tex] and I understand that the answer will probably be an approximation. I'm hoping for an exact solution though, however ugly it may be.

    Any ideas?

    p.s. this isn't a homework problem, so it may well be that you can't have the function be zero for all x between 0 to 1 for any n. It seems as though for some n I've chosen, the function is always going to be under the x axis or above the x axis depending on my n. If it so happens that this is always the case even for n approaching very close to my desired value, then my question should have a valid solution. I'm curious as to how I could show it is always above/below the axis for some n or if it isn't.
     
    Last edited: Aug 30, 2010
  2. jcsd
  3. Aug 30, 2010 #2

    Mentallic

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    Oh even for n=2.5 it cuts the axis in between 0 and 1, what a let down... in that case, I withdraw my question.
     
  4. Aug 31, 2010 #3

    jgens

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    Gold Member

    I realize that you've withdrawn this, but you should be able to prove that there is no number n satisfying the desired conditions (assuming that I've interpretted your question correctly).

    Define the function [itex]f[/itex] such that [itex]f(x)=\log{(\cos{(x)})}-x^n[/itex] and suppose that there is some number [itex]n[/itex] such that [itex]f(x)=0[/itex] for all [itex]x \in [0,1][/itex]. The last condition means that [itex]f[/itex] is constant on an interval; and moreover, [itex]f[/itex] is differentiable on this interval too. Therefore, [itex]f'(x)=-(\tan{(x)}+nx^{n-1})=0[/itex] for all [itex]x \in [0,1][/itex]. It follows from the last equality that [itex]nx^{n-1} = -\tan{(x)}[/itex] for [itex]x \in [0,1][/itex], but this is a contradiction since [itex]nx^{n-1} \geq 0[/itex] and [itex]\tan{(x)} \geq 0[/itex] on the desired interval.* This essentially completes the proof.**

    I'm sorry if there's some fundamental error in the 'proof' above or if I misunderstood your question. Hopefully you'll find something in the above post useful.

    * I'm assuming [itex]n>0[/itex], because otherwise [itex]f[/itex] isn't defined for [itex]x=0[/itex].

    ** I used a modification of your function, but the proof for the function that you gave should be analogous to this one (assuming that I didn't screw up somewhere).
     
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