# Extrema can be Easy!

Calculus Level 5

Let $$f(n)$$ and $$g(k)$$ be functions from the positive integers to the reals such that $f(n)=\sum _{ x=1 }^{ n }{ \sin ^{ 5 }{ (x^{\circ}) } }$ $g(k) = \sum _{ x=1 }^{ k }{ \cos ^{ 7 }{ (x^{\circ}) } }$ Let the $$\text{Maximum Value}$$ of $$f(n) - g(k)$$ be $$M$$. Given that $$a^2+b^2+c^2=\lfloor M\rfloor$$, what is the $$\text{Maximum Value}$$ of $\dfrac{2ab}{b-a^2b}-\dfrac{-4b^2 - 4b}{-2b^3 -2b^2 +2b + 2}-\dfrac{c^2(10c+10)}{5c^4+5c^3-5c(c+1)}$ $$\text{for } a, b, c\ge 2$$?

×