In this problem, we shall prove that iff$($x$)$ and g$($x$)$ are suitably behaved functions, then f$($x$)$f$($x$)$ dx g$($x$)$g$($x$)$ dx $$

$14$

$($f$($x$)$g$($x$)+$ f$($x$)$g$($x$))$ dx

We shall use this inequality to derive Equation $4\mathrm{.}19$ in the next problem. First, let A$=$

f$($x$)$f$($x$)$ dx B $=$ f$($x$)$g$($x$)$ dx C $=$g$($x$)$g$($x$)$ dx

Now argue that

$[$f$($x$)+$ g$($x$)][$f$($x$)+$ g$($x$)]$ dx $=$ A$2+($B $+$ B$)$ $+$ C

is equal to or greater than zero for real values of $.$ Show that A$>0,$ C$>0,$ and that B $+$ B$0,$ and then argue that the roots of the quadratic form A$2+($B $+$ B$)$ $+$ C cannot be real. Show that this can be so only if

AC $($B $+$ B$)2$

$14$

which is the same as the above inequality.