Q102 / 160 · AP EAMCET 2024 (21 May Shift 2) · Mathematics
If $3 f(x)-2 f\left(\frac{1}{x}\right)=x$ then $f^{\prime}(2)=$
- A1
- Correct answerB$\frac{1}{2}$
- C2
- D$\frac{7}{2}$
Solution
$3 f(x)-2 f\left(\frac{1}{x}\right)=x$...(i)
Replacing $x$ by $\frac{1}{x}$, we get
$3 f\left(\frac{1}{x}\right)-2 f(x)=\frac{1}{x}$...(ii)
Solving (i) and (ii), $f(x)=\frac{1}{5}\left(3 x+\frac{2}{x}\right)$
$\Rightarrow f^{\prime}(x)=\frac{3}{5}-\frac{2}{5 x^2} \Rightarrow f^{\prime}(2)=\frac{3}{5}-\frac{1}{10}=\frac{1}{2}$.
Replacing $x$ by $\frac{1}{x}$, we get
$3 f\left(\frac{1}{x}\right)-2 f(x)=\frac{1}{x}$...(ii)
Solving (i) and (ii), $f(x)=\frac{1}{5}\left(3 x+\frac{2}{x}\right)$
$\Rightarrow f^{\prime}(x)=\frac{3}{5}-\frac{2}{5 x^2} \Rightarrow f^{\prime}(2)=\frac{3}{5}-\frac{1}{10}=\frac{1}{2}$.
Hence, the correct answer is option: 'B'.