Question

Let f(x) and g(x) be two functions having finite non-zero third order derivatives $f^{{}^{\prime }"}\left(x\right)$ and $g^{{}^{\prime }"}\left(x\right)$ or all $xϵR$. If $f\left(x\right)g\left(x\right)=1forallxϵR$, then $\frac{f^{{}^{\prime }"}}{f^{{}^{\prime }}}-\frac{g^{{}^{\prime }"}}{g^{{}^{\prime }}}$ is equal to:

• A. $3\left(\frac{f^{\prime \prime }}{g}\right)-\left(\frac{g^{\prime \prime }}{f}\right)$
• B. $3\left(\frac{f^{\prime \prime }}{f}\right)-\left(\frac{g^{\prime \prime }}{g}\right)$
• C. $3\left(\frac{g^{\prime \prime }}{g}\right)-\left(\frac{f^{\prime \prime }}{g}\right)$
• D. $3\left(\frac{f^{\prime \prime }}{g}\right)-\left(\frac{g^{\prime \prime }}{f}\right)$

Answer 1

The correct option is B $3\left(\frac{f^{\prime \prime }}{f}\right)-\left(\frac{g^{\prime \prime }}{g}\right)$

$fg=1\Rightarrow f{g}_1+f_{1}g=0\Rightarrow f{g}_2+g{f}_2+2f_1{g}_1=0\Rightarrow f{g}_3+g{f}_3=-3[f_1{g}_2+f_2{g}_1]\Rightarrow \frac{f_3}{f_1}-\frac{g_3}{g_1}=3[\frac{f_2}{f}-\frac{g_2}{g}](usingf{g}_1=-f_{1}g)$