Question

Suppose f and g are functions having second derivatives f'' and g'' every where, if f(x).g(x)=1 for all x and f' and g' are never zero, then $\frac{f^{\prime \prime }\left(x\right)}{f^{\prime }\left(x\right)}-\frac{g^{\prime \prime }\left(x\right)}{g^{\prime }\left(x\right)}$ equals

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$Wehave{f}^{\prime }\left(x\right)g\left(x\right)+g^{\prime }\left(x\right)f\left(x\right)=0\Rightarrow \frac{g\left(x\right)}{g^{\prime }\left(x\right)}+\frac{f\left(x\right)}{f^{\prime }\left(x\right)}---\left(1\right)Further{f}^{\prime \prime }\left(x\right)g\left(x\right)+2f^{\prime }\left(x\right)g^{\prime }\left(x\right)+g^{\prime \prime }\left(x\right)f\left(x\right)=0Dividethroughoutby{f}^{\prime }\left(x\right)g^{\prime }\left(x\right)anduse\left(1\right)$