Explanation

(c)
$\frac{d}{dx}\left\lbrack f^{2}(x) + g^{2}(x) \right\rbrack$
$= 2\left\lbrack f(x)f^{'}(x) + g(x)g^{'}(x) \right\rbrack$
$= 2\left\lbrack f(x)g(x) + g(x)\left\{ - f(x) \right\} \right\rbrack$
= 0
$\Rightarrow f^{2}(x) + g^{2}(x) = constant$
$\therefore\ \ f^{2}(4) + g^{2}(4) = f^{2}(2) + g^{2}(2)$
$= (4)^{2} + (4)^{2}$
=$32$