If $$\alpha $$ $$ = 3{\sin ^{ - 1}}\left( {{6 \over {11}}} \right)$$ and $$\beta = 3{\cos ^{ - 1}}\left( {{4 \over 9}} \right),$$ where the inverse trigonimetric functions take only the principal values, then the correct options(s) is (are)

Let $$f\left( x \right) = 7{\tan ^8}x + 7{\tan ^6}x - 3{\tan ^4}x - 3{\tan ^2}x$$ for all $$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right).$$
Then the correct expression(s) is (are)

The option(s) with the values of a and $$L$$ that satisfy the following equation is (are) $$${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} } \over {\int\limits_0^\pi {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} }} = L?$$$

Let $$f'\left( x \right) = {{192{x^3}} \over {2 + {{\sin }^4}\,\pi x}}$$ for all $$x \in R\,\,$$ with $$f\left( {{1 \over 2}} \right) = 0$$.
If $$m \le \int\limits_{1/2}^1 {f\left( x \right)dx \le M,} $$ then the possible values of $$m$$ and $$M$$ are