Extremas of a Function!

Calculus Level 5

\[\large{f(\cot(x)) = \sin(2x) + \cos(2x)}\]

Let \(f\) be a function defined on the set of real numbers \(\mathbb R\), taking the values in \(\mathbb R\), and satisfying the above condition for \(\forall \ x \in (0,\pi) \).

If the sum of the least and greatest values of the function \(g(x) = f(x) \cdot f(1-x)\) on the closed interval \([-1,1]\) can be expressed as:

\[\large{\dfrac{A}{B} - \sqrt{C}}\]

such that \(A,B,C \in \mathbb Z^+\ ; \ \gcd(A,B)=1\) and \(C\) has no perfect \(n^{th}\) power factor, \(n \in \mathbb Z^+, n \geq 2\). Submit the value of \(A+B+C\) as your answer.


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