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 $(0, \pi/2)$

(B) $f(x)$ is decreasing on $(\pi/2, \pi)$

(C) $f(x)$ is increasing on $(0, \pi)$

(D) $f(x)$ is decreasing on $(0, \pi)$

Choose the correct answer from the options given below:

Given $f(x) = \log_e(\sin x)$ where $x \in (0, \pi)$.

To determine if the function is increasing or decreasing, find the derivative and check its sign.

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Using the chain rule:

$f(x) = \log_e(\sin x)$

$f'(x) = \frac{1}{\sin x} \times \frac{d}{dx}(\sin x)$

$f'(x) = \frac{1}{\sin x} \times \cos x$

$f'(x) = \frac{\cos x}{\sin x}$

$f'(x) = \cot x$

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For statement (A), checking on $(0, \pi/2)$:

In the interval $(0, \pi/2)$:

• $\sin x > 0$
• $\cos x > 0$
• Therefore, $\cot x = \frac{\cos x}{\sin x} > 0$

Since $f'(x) > 0$, the function is increasing on $(0, \pi/2)$.

Statement (A) is TRUE.

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For statement (B), checking on $(\pi/2, \pi)$:

In the interval $(\pi/2, \pi)$:

• $\sin x > 0$
• $\cos x < 0$
• Therefore, $\cot x = \frac{\cos x}{\sin x} < 0$

Since $f'(x) < 0$, the function is decreasing on $(\pi/2, \pi)$.

Statement (B) is TRUE.

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For statements (C) and (D), checking on $(0, \pi)$:

At $x = \pi/2$, the derivative changes sign from positive to negative.

The function increases on $(0, \pi/2)$ and decreases on $(\pi/2, \pi)$.

Therefore, the function is neither completely increasing nor completely decreasing on $(0, \pi)$.

Statement (C) is FALSE.

Statement (D) is FALSE.

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Only statements (A) and (B) are TRUE.

The correct answer is: (A) and (B) only.