Explanation

(c)
We have, $8f(x) + 6f\left( \frac{1}{x} \right)$=$x$+5 for all $x$ …(i)
Therefore, $8f\left( \frac{1}{x} \right) + 6f(x) = \frac{1}{x}$+5 …(ii)
From Eqs. (i) and (ii), we have
$f(x) = \frac{1}{28}\left( 8x - \frac{6}{x} + 10 \right)$
Now, $y = x^{2}\ f(x)$
$\ \Rightarrow \ \ \ \ \ \ \ \ y = \frac{1}{28}(8x^{2} - 6x + 10x^{2})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \frac{dy}{dx} = \frac{1}{28}(24x^{2} + 20x - 6)$
$\therefore\ \ \left( \frac{dy}{dx} \right)_{x = - 1} = \frac{1}{28}(24 - 20 - 6)$
$= - \frac{1}{14}$