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

Question

Let f ( x ) be a function such that f '( x )= log 1 / 3[ log 3( sin x + a )]. If f ( x ) is decreasing for all real values of x, then (A) a ∈(1,4) (B) a ∈(4, ∞) (C) a ∈(2,3) (D) a ∈(2, ∞)

Tardigrade Admin · Accepted Answer

We must have log 1 / 3( log 3( sin x+a))<0 ∀ x ∈ R ⇒ log 3( sin x+a)>1 ∀ x ∈ R ⇒ sin x+a>3 ∀ x ∈ R ⇒ a>3- sin x ∀ x ∈ R ⇒ a>4

Q. Let $f (x)$ be a function such that $f^{'} (x) = lo g_{1/3} [lo g_{3} (sin x + a)]$ . If $f (x)$ is decreasing for all real values of $x$ , then

$a \in (1, 4)$

$a \in (4, \infty)$

$a \in (2, 3)$

$a \in (2, \infty)$

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

Q. Let f(x) be a function such that f′(x)=log1/3​[log3​(sinx+a)]. If f(x) is decreasing for all real values of x, then

a∈(1,4)

a∈(4,∞)

a∈(2,3)

a∈(2,∞)