QJEEAdv15Mac9d4: JEE Advanced 2015 Mathematics - Class 12 JEEAdv15Mac9d4 Solution: Let for all with . If then the possible values of and are

We have, $f'(x) = {{192{x^3}} \over {2 + {{\sin }^4}\pi x}}$

Given, $f\left( {{1 \over 2}} \right) = 0$ and $m \le \int\limits_{1/2}^1 {f(x)dx \le m}$

$\therefore$ f(x) is increasing in $\left( {{1 \over 2},1} \right)$ .

$\therefore$ $f'{(x)_{\max }} = {{192} \over {24}} = 96$

$\Rightarrow 96 = {{f(1) - f(1/2)} \over {1/2}} \Rightarrow f(1) = 96 \times {1 \over 2} = 48$

$M = {1 \over 2} \times {1 \over 2} \times 48 = 12$

$f'{(x)_{\min .}} = {{\left( {{{192} \over 8}} \right)} \over 3} = {{192} \over {24}}$

$\Rightarrow {{192} \over {24}} = {{f(1) - 0} \over {(1/2)}} \Rightarrow f(1) = {{192} \over {24}} \times {1 \over 2}$

$\therefore$ $m = {1 \over 2} \times {1 \over 2} \times {{192} \over {24}} \times {1 \over 2} = 1$