Question

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

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

Answer 1

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

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