Question

If $5f\left(x\right)+3f\left(\frac{1}{x}\right)=x+2$ and $y=xf\left(x\right),then(\frac{dy}{dx}{)}_{x=1}$ is equal to

• A. $14$
• B. $\frac{7}{8}$
• C. $1$
• D. $Noneofthese$

Answer 1

The correct option is B $\frac{7}{8}$

$5f\left(x\right)+3f\left(\frac{1}{x}\right)=x+2...\left(i\right)$
$Replacingxby\frac{1}{x}$
$\therefore 5f\left(\frac{1}{x}\right)+3f\left(x\right)=\frac{1}{x}+2....\left(ii\right)$
From Eq. (i),
$25f\left(x\right)+15f\left(\frac{1}{x}\right)=5x+10....\left(iii\right)$
and from Eq. (ii)
$9f\left(x\right)+15f\left(\frac{1}{x}\right)=\frac{3}{x}+6....\left(iv\right)$
Subtracting Eq. (iv) from (iii), we get
$\therefore 16f\left(x\right)=5x-\frac{3}{x}+4\therefore xf\left(x\right)=\frac{5x^2-3+4x}{16}=y\therefore \frac{dy}{dx}=\frac{10x+4}{16}\therefore (\frac{dy}{dx}{)}_{x=1}$ = \frac{10+4}{16}\\
= \frac{7}{8}\)