Find n for y=ln(cos(x))/cos(1)-x^n to be Zero

[Mentallic]

Homework Statement

I need to find n such that,

y=\frac{ln(cos(x))}{cos(1)}-x^n

is zero for all 0\leq x\leq 1.

The Attempt at a Solution

I've already narrowed it down to 2<n<2.5 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.

 

[Mentallic]

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.

 

[jgens]

I realize that you've withdrawn this, but you should be able to prove that there is no number n satisfying the desired conditions (assuming that I've interpretted your question correctly).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 f such that f(x)=\log{(\cos{(x)})}-x^n and suppose that there is some number n such that f(x)=0 for all x \in [0,1]. The last condition means that f is constant on an interval; and moreover, f is differentiable on this interval too. Therefore, f'(x)=-(\tan{(x)}+nx^{n-1})=0 for all x \in [0,1]. It follows from the last equality that nx^{n-1} = -\tan{(x)} for x \in [0,1], but this is a contradiction since nx^{n-1} \geq 0 and \tan{(x)} \geq 0 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 n>0, because otherwise f isn't defined for x=0.

** 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).