For non-zero $x$, if $cf(x)+df(1/x)=|\log|x||+3$ where $c \ne d$, then $f(x)$ equals
- A. $(c-d)(2e-1)/(c^2-d^2)$
- B. $(c-d)(3e-2)/(c^2-d^2)$
- C. $(c-d)(3e+2)/(c^2-d^2)$
- D. $(c-d)(2e+1)/(c^2-d^2)$
✅ Correct Answer: $(c-d)(3e-2)/(c^2-d^2)$
Explanation
Given $cf(x)+df(1/x)=|\log|x||+3$ Replacing $x$ by $1/x$ $df(x)+cf(1/x)=|\log|x||+3$ Multiply first equation by $c$ and second by $d$ and subtract $(c^2-d^2)f(x)=(c-d)(|\log|x||+3)$ For $1<x<e$, $|\log|x||=\log x$ $f(x)=(c-d)(\log x+3)/(c^2-d^2)$
🎯 Happy Preparation — ACME Academy
