Let \(f(x)= \log_e (\sin x), x \in (0,\pi)\), then which of the following statements is/are TRUE?
(A) f(x) is increasing on (0, π/2)
(B) f(x) is decreasing on (π/2, π)
(C) f(x) is increasing on (0, π)
(D) f(x) is decreasing on (0, π)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
By the chain rule,
\( \displaystyle f'(x) = \frac{1}{\sin x}\cdot\cos x \)
\(f'(x) = \cot x\)
The sign of the derivative decides where f increases or decreases:
\( \displaystyle \cot x \gt 0 \text{ for } 0 \lt x \lt \frac{\pi}{2} \)
\( \displaystyle \cot x \lt 0 \text{ for } \frac{\pi}{2} \lt x \lt \pi \)
So f is increasing on \(\left(0, \frac{\pi}{2}\right)\) and decreasing on \(\left(\frac{\pi}{2}, \pi\right)\).
Checking the statements
(A) f(x) is increasing on (0, π/2) — Correct. \(f'(x) = \cot x \gt 0\) there.
(B) f(x) is decreasing on (π/2, π) — Correct. \(f'(x) = \cot x \lt 0\) there.
(C) f(x) is increasing on (0, π) — Incorrect. f decreases on \(\left(\frac{\pi}{2}, \pi\right)\).
(D) f(x) is decreasing on (0, π) — Incorrect. f increases on \(\left(0, \frac{\pi}{2}\right)\).
NCERT (2026-27 edition):
Increasing and Decreasing Functions
