Question

Let $f$ and $g$ be two differentiable functions defined from $R \rightarrow R^{+}$. If $f(x)$ has a local maximum at $x=c$ and $g(x)$ has a local minimum at $x=c$, then $h(x)=\frac{f(x)}{g(x)}$

• A. has a local maximum at $x = c$
• B. has a local minimum at $x = c$
• C. is monotonic at $x = c$
• D. has a point of inflection at $x = c$

Answer 1

Answer: (A)

Given, $h(x)=\frac{f(x)}{g(x)}$
$\Rightarrow h ^{\prime}( x )=\frac{ g ( x ) f ^{\prime}( x )- f ( x ) g ^{\prime}( x )}{ g ^2( x )}$
Clearly, $h^{\prime}( c )=0 \left(\right.$ As, $f ^{\prime}( c )=0$ and $\left.g ^{\prime}( c )=0\right)$
$\text { So, } h ^{\prime}\left( c ^{-}\right)=\frac{ g \left( c ^{-}\right) f ^{\prime}\left( c ^{-}\right)- f \left( c ^{-}\right) \cdot g ^{\prime}\left( c ^{-}\right)}{ g ^2\left( c ^{-}\right)} \left[ f ^{\prime}\left( c ^{-}\right)>0 ; g ^{\prime}\left( c ^{-}\right)<0 ; f \left( c ^{-}\right)>0 ; g \left( c ^{-}\right)>0\right] $
$\therefore h ^{\prime}\left( c ^{-}\right)>0 $
$|||^{l y} h ^{\prime}\left( c ^{+}\right)<0$
So, $ h(x)$ has a local maximum at $x=c$.