Which of the following functions are increasing on $x ∈(0,\frac{\pi}{2})$?
(A) $f(x) = \sin x$
(B) $f(x) = \cos x$
(C) $f(x) = \tan x$
(D) $f(x) = \cos 3x$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
(A) $f(x) = \sin x$
(C) $f(x) = \tan x$
Given interval: $x \in (0, \frac{\pi}{2})$
Check monotonicity using derivative:
(A) $f(x) = \sin x$
$f'(x) = \cos x > 0$ in $(0, \frac{\pi}{2})$
⇒ Increasing
(B) $f(x) = \cos x$
$f'(x) = -\sin x < 0$ in $(0, \frac{\pi}{2})$
⇒ Decreasing
(C) $f(x) = \tan x$
$f'(x) = \sec^2 x > 0$ in $(0, \frac{\pi}{2})$
⇒ Increasing
(D) $f(x) = \cos 3x$
$f'(x) = -3\sin 3x < 0$ in $(0, \frac{\pi}{2})$
⇒ Decreasing