(i)     Suppose that $a$, $b$ and $t$ are positive. Which of the following two expressions is the larger

${\displaystyle P={\left({\int }_0^t{x}^{a+b}\mathrm{dx}\right)}^2,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}Q=\left({\int }_0^t{x}^{2a}\mathrm{dx}\right)\left({\int }_0^t{x}^{2b}\mathrm{dx}\right)\hspace{0.5em}?}$

(ii)     By considering the inequality

${\displaystyle {\int }_0^t[f(x)+\lambda g(x{)]}^2\mathrm{dx}\ge 0,}$

prove that, for all functions $f(x)$ and $g(x)$,

${\displaystyle {({\int }_0^{t}f(x)g(x)\mathrm{dx})}^2\le ({\int }_0^{t}f(x{)}^2\mathrm{dx})({\int }_0^{t}g(x{)}^2\mathrm{dx}).}$