Question

Let $f\left(x\right)$ be a function such that $f^{\prime }\left(x\right)={\log }_{\frac{1}{3}}[{\log }_3\sin x+a)]$. if $f\left(x\right)$ is decreasing for all real values of $x$, then

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

Answer 1

The correct option is C $a\in (4,\infty )$

$f\left(x\right)$ is decreasing $\Rightarrow f^{\prime }\left(x\right)<0$

$\Rightarrow {\log }_{1/3}\left[{\log }_3\right(\sin x+a\left)\right]<0\Rightarrow -{\log }_3\left[{\log }_3\right(\sin x+a\left)\right]<0\Rightarrow {\log }_3\left[{\log }_3\right(\sin x+a\left)\right]>0\Rightarrow {\log }_3(\sin x+a)>1\Rightarrow a+\sin x>3\Rightarrow a>3-\left(\sin x\right)\Rightarrow a>3-[-1,1]\Rightarrow a>4$

Hence, correct answer is $a\in (4,\infty )$