Question

If $F\left(x\right)=f\left(x\right).g\left(x\right)and{f}^{\prime }\left(x\right)g^{\prime }\left(x\right)=c$ , where ‘c’ is a constant then $\frac{f‘‘}{f}+\frac{g‘‘}{g}+\frac{2C}{fg}$=

• A. $\frac{f^{\prime \prime }}{F}$
• B. $\frac{f^{\prime \prime }}{F^{\prime }}$
• C. $\frac{F^{\prime \prime }}{F}$
• D. $Noneofthese$

Answer 1

The correct option is C $\frac{F^{\prime \prime }}{F}$

$GivenF=fg\Rightarrow F^{\prime }=f^{\prime }g+f{g}^{\prime }{F}^{\prime \prime }=f^{\prime \prime }g+f^{\prime }{g}^{\prime }+f^{\prime }{g}^{\prime }+f{g}^{\prime \prime }\Rightarrow F^{\prime \prime }=f^{\prime \prime }g+2C+f{g}^{\prime \prime }\Rightarrow \frac{F^{\prime \prime }}{F}=\frac{f^{\prime \prime }}{f}+\frac{g^{\prime \prime }}{g}+\frac{2C}{fg}$