Upon heating $$KCl{O_3}$$ in the presence of catalytic amount of $$Mn{O_2},$$ a gas $$W$$ is formed. Excess amount of $$W$$ reacts with white phosphorous to give $$X.$$ The reaction of $$X$$ with pure $$HN{O_3}$$ gives $$Y$$ and $$Z.$$

$$W$$ and $$X$$ are, respectively

The reaction of compound $$P$$ with $$C{H_3}MgBr$$ (excess) in $${\left( {{C_2}{H_5}} \right)_2}O$$ followed by addition of $${H_2}O$$ gives $$Q.$$ The compound $$Q$$ on treatment with $${H_2}S{O_4}$$ at $${0^ \circ }C$$ gives $$R.$$ The reaction of $$R$$ with $$C{H_3}COCl$$ in the presence of anhydrous $$AlC{l_3}$$ in $$C{H_2}C{l_2}$$ followed by treatment with $${H_2}O$$ producess compound $$S.$$ [$$Et$$ in compound $$P$$ is ethyl group]

The reactions, $$Q$$ to $$R$$ and $$R$$ to $$S,$$ are

The major product of the following reaction is

If f : R $$ \to $$ R is a twice differentiable function such that f"(x) > 0 for all x$$ \in $$R, and $$f\left( {{1 \over 2}} \right) = {1 \over 2}$$, f(1) = 1, then