# 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.

×