Search any question & find its solution
Question:
Answered & Verified by Expert
If \(3 f(x)-2 f\left(\frac{1}{x}\right)=x\), then \(f^{\prime}(2)\) is
Options:
Solution:
1695 Upvotes
Verified Answer
The correct answer is:
\(\frac{1}{2}\)
\(3 f(x)-2 f\left(\frac{1}{x}\right)=x, f^{\prime}(2)=\) ?
\(\Rightarrow 3 f^{\prime}(x)-2 f^{\prime}\left(\frac{1}{x}\right)\left(-\frac{1}{x^2}\right)=1\)
At \(\quad(x=2) \Rightarrow 3 f^{\prime}(2)+\frac{1}{2} f^{\prime}\left(\frac{1}{2}\right)=1\) ...(i)
And At \(\left(x=\frac{1}{2}\right) \Rightarrow 3 f^{\prime}\left(\frac{1}{2}\right)+8 f^{\prime}(2)=1\) ...(ii)
On solving Eqs. (i) and (ii), we get
\(\Rightarrow \quad\left\{f^{\prime}(2)=\frac{1}{2}\right\}\)
\(\Rightarrow 3 f^{\prime}(x)-2 f^{\prime}\left(\frac{1}{x}\right)\left(-\frac{1}{x^2}\right)=1\)
At \(\quad(x=2) \Rightarrow 3 f^{\prime}(2)+\frac{1}{2} f^{\prime}\left(\frac{1}{2}\right)=1\) ...(i)
And At \(\left(x=\frac{1}{2}\right) \Rightarrow 3 f^{\prime}\left(\frac{1}{2}\right)+8 f^{\prime}(2)=1\) ...(ii)
On solving Eqs. (i) and (ii), we get
\(\Rightarrow \quad\left\{f^{\prime}(2)=\frac{1}{2}\right\}\)
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.