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  4. If y=x2 f(x), 8 f(x)+6 f((1/x))=x+5 and ((d y/d x))x=-1=(1/a), then find the value of a.

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Q. If $y=x^{2} f(x), 8 f(x)+6 f\left(\frac{1}{x}\right)=x+5$ and $\left(\frac{d y}{d x}\right)_{x=-1}=\frac{1}{a}$, then find the value of $a$.

Limits and Derivatives

Solution:

$8 f(x)+6 f\left(\frac{1}{x}\right)=x+5$ ...(i)
Replacing $x$ by $\frac{1}{x}$, we get
$8 f\left(\frac{1}{x}\right)+6 f(x)=\frac{1}{x}+5$ ...(ii)
Solving (i.) and (ii), we get
$f(x) =\frac{1}{14}\left(4 x-\frac{3}{x}+5\right)$
$y =x^{2} f(x)$
$=x^{2}\left[\frac{1}{14}\left(4 x-\frac{3}{x}+5\right)\right]$
$\Rightarrow y=\frac{1}{14}\left(4 x^{3}-3 x+5 x^{2}\right)$
$\Rightarrow \frac{d y}{d x}=\frac{1}{14}\left(12 x^{2}-3+10 x\right)$
$\Rightarrow\left(\frac{d y}{d x}\right)_{x=-1}=\frac{1}{14}(12-3-10)=\frac{-1}{14}$
$\Rightarrow a=-14$