Extrema can be Easy!

Calculus Level 5

Let f(n)f(n) and g(k)g(k) be functions from the positive integers to the reals such that f(n)=x=1nsin5(x)f(n)=\sum _{ x=1 }^{ n }{ \sin ^{ 5 }{ (x^{\circ}) } } g(k)=x=1kcos7(x)g(k) = \sum _{ x=1 }^{ k }{ \cos ^{ 7 }{ (x^{\circ}) } } Let the Maximum Value\text{Maximum Value} of f(n)g(k)f(n) - g(k) be MM. Given that a2+b2+c2=Ma^2+b^2+c^2=\lfloor M\rfloor, what is the Maximum Value\text{Maximum Value} of 2abba2b4b24b2b32b2+2b+2c2(10c+10)5c4+5c35c(c+1)\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)} for a,b,c2\text{for } a, b, c\ge 2?

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