Question

Let f(x) be a function such that $f\text{'}\left(x\right)={\log }_{1/3}\left({\log }_3\left(\sin x+a\right)\right]$. If f(x) is decreasing for all real values of x, then .

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

Answer 1

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

Given that,$f\text{'}\left(x\right)={\log }_{1/3}\left({\log }_3\left(\sin x+a\right)\right]$
Since, f(x) is a decreasing function for all the values of a, then
$f\text{'}\left(x\right)<0\forall x\in R\Rightarrow {\log }_{1/3}\left({\log }_3\left(\sin x+a\right)\right]<0,\forall x\in R\Rightarrow {\log }_3\left(\sin x+a\right)>1,\forall x\in R\Rightarrow \sin \left(x\right)+a>3,\forall x\in R\Rightarrow a>3-\sin x,\forall x\in R\Rightarrow a>4\Rightarrow a\in \left(4,\infty \right).$