Cosine and Something Else

Geometry Level 5

Let \(n\) be a positive nonzero integer and
\(k=4n,\)
\(a=x+\sqrt { { x }^{ 2 }-{ k }^{ 2 } }, \)
\(b=x-\sqrt { { x }^{ 2 }-{ k }^{ 2 } }, \)
\(f\left( x \right) =\dfrac { 1 }{ 2 } \dfrac { 1 }{ { k }^{\displaystyle k } } \left( { a }^{ \displaystyle k }+{ b }^{\displaystyle k } \right).\)

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

You may want to use a graphing calculator.

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