The range of the function f (θ)=( sin θ/θ)+(θ/ tan θ), θ ∈(0, (π/2)) is equal to

Question

The range of the function f (θ)=( sin θ/θ)+(θ/ tan θ), θ ∈(0, (π/2)) is equal to (A) (0, ∞) (B) ((1/π), 2) (C) (2, ∞) (D) ((2/π), 2)

Tardigrade Admin · Accepted Answer

We know that ( sin θ/θ) and (θ/ tan θ) both are decreasing functions of θ in (0, (π/2)). So maximum value, when θ arrow 0 is 1+1=2 and minimum value, when θ arrow (π/2) is (2/π).

Q. The range of the function $f (θ) = \frac{s i n θ}{θ} + \frac{θ}{t a n θ}, θ \in (0, \frac{π}{2})$ is equal to

$(0, \infty)$

$(\frac{1}{π}, 2)$

$(2, \infty)$

$(\frac{2}{π}, 2)$

We know that
$\frac{s i n θ}{θ}$ and $\frac{θ}{t a n θ}$ both are decreasing functions of $θ$ in $(0, \frac{π}{2})$ . So maximum value, when $θ \to 0$ is
$1 + 1 = 2$ and minimum value, when $θ \to \frac{π}{2}$ is $\frac{2}{π}$ .

Q. The range of the function f(θ)=θsinθ​+tanθθ​,θ∈(0,2π​) is equal to

(0,∞)

(π1​,2)

(2,∞)

(π2​,2)