Cosine and Something Else

Geometry Level 5

Let nn be a positive nonzero integer and

k=4n,k=4n,
a=x+x2k2,a=x+\sqrt { { x }^{ 2 }-{ k }^{ 2 } },
b=xx2k2,b=x-\sqrt { { x }^{ 2 }-{ k }^{ 2 } },
f(x)=121kk(ak+bk).f\left( x \right) =\dfrac { 1 }{ 2 } \dfrac { 1 }{ { k }^{\displaystyle k } } \left( { a }^{ \displaystyle k }+{ b }^{\displaystyle k } \right).

For what integer n<1000n<1000 does f(x)f\left( x \right) intersect cos(x)\cos\left( x \right) the most times between 4π<x<4π-4\pi <x<4\pi ?

You may want to use a graphing calculator.

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