Question

Let $f ( x )$ be a function such that $f '( x )=\log _{1 / 3}\left[\log _{3}(\sin x + a )\right]$. If $f ( x )$ is decreasing for all real values of $x$, then

• A. $a \in(1,4)$
• B. $a \in(4, \infty)$
• C. $a \in(2,3)$
• D. $a \in(2, \infty)$

Answer 1

Answer: (B)

We must have $\log _{1 / 3}\left(\log _{3}(\sin x+a)\right)<0 \forall x \in R$
$\Rightarrow \log _{3}(\sin x+a)>1 \forall x \in R$
$\Rightarrow \sin x+a>3 \forall x \in R$
$\Rightarrow a>3-\sin x \forall x \in R $
$\Rightarrow a>4$