# 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.

×