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\[\] \[\dfrac{2ab}{b-a^2b}-\dfrac{-4b^2 - 4b}{-2b^3 -2b^2 +2b + 2}-\dfrac{c^2(10c+10)}{5c^4+5c^3-5c(c+1)}\]\[\] \(\text{for } a, b, c\ge 2\)?\[\]

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