Question

Let $f\left(x\right)$ be a function such that $f^{\prime }\left(x\right)={\log }_{\frac{1}{3}}\left[{\log }_3\right(\sin x+a\left)\right].$
If $f\left(x\right)$ 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

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

$f\left(x\right)$ is such that $f^{\prime }\left(x\right)={log}_{1/3}\left[{log}_3(sinx+a)\right]$

Now, if $f\left(x\right)$ is decreasing for all real values of $x$

$\Rightarrow f^{\prime }\left(x\right)=0$

$\therefore {log}_{1/3}\left[{log}_3(sinx+a)\right]<{log}_{1/3}\left(1\right)$

Since base of log is $<0$

$\Rightarrow {log}_3(sinx+a)>1\Rightarrow {log}_3(sinx+a)>{log}_3\left(3\right)\Rightarrow (sinx+a)>3\Rightarrow sinx>3-aSincesinx\ge -1\Rightarrow 3-a\ge -1\Rightarrow a\ge 4\therefore a\in [4,\infty ]$