Consider two statements $S_1$ and $S_2$. $S_1$: If $f(x)$ is a differentiable function with $f^{\prime}(x) = 0$ in $(a, b)$ and $f(x)$ is increasing in $(a, b)$,then $\frac{f(x)}{f^{\prime}(x)}$ is also increasing in $(a, b)$. $S_2$: Both $\sin x$ and $\tan x$ are increasing functions in $\left(0, \frac{\pi}{2}\right)$. Which of the following is true?

• A
  Both $S_1$ and $S_2$ are wrong.
• B
  $S_1$ is correct and implies $S_2$.
• C
  $S_1$ is wrong and $S_2$ is right.
• D
  Both $S_1$ and $S_2$ are right.