Consider the function $f(x) = \sin x$ in the interval $[\pi, 2\pi]$ then which of the following statements are correct?

(A) $x = \frac{3\pi}{2}$ is its stationary point.

(B) Its maximum value is 1

(C) Its minimum value is -1

(D) It attains its maximum value at $\pi$ and $2\pi$

Choose the correct answer from the options given below:

The function $f(x) = \sin x$ is analyzed in the interval $[\pi, 2\pi]$.

Understanding the behavior in this interval:

At $x = \pi$: $\sin(\pi) = 0$

Between $\pi$ and $\frac{3\pi}{2}$: $\sin x$ is negative

At $x = \frac{3\pi}{2}$: $\sin\left(\frac{3\pi}{2}\right) = -1$ (lowest point)

Between $\frac{3\pi}{2}$ and $2\pi$: $\sin x$ increases back toward 0

At $x = 2\pi$: $\sin(2\pi) = 0$

In the interval $[\pi, 2\pi]$, the sine wave is entirely non-positive.

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Statement (A): $x = \frac{3\pi}{2}$ is a stationary point

A stationary point occurs where the derivative equals zero.

$f'(x) = \cos x$

At $x = \frac{3\pi}{2}$:

$f'\left(\frac{3\pi}{2}\right) = \cos\left(\frac{3\pi}{2}\right) = 0$

Statement (A) is correct.

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Statement (B): Its maximum value is 1

The function values in $[\pi, 2\pi]$ range from $0$ to $-1$ and back to $0$.

The highest value is $0$ at the endpoints.

The value never reaches $1$ in this interval.

Statement (B) is incorrect.

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Statement (C): Its minimum value is -1

At $x = \frac{3\pi}{2}$:

$\sin\left(\frac{3\pi}{2}\right) = -1$

This is the lowest point in the interval.

Statement (C) is correct.

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Statement (D): It attains its maximum value at $\pi$ and $2\pi$

The maximum value in $[\pi, 2\pi]$ is $0$.

At $x = \pi$: $\sin(\pi) = 0$

At $x = 2\pi$: $\sin(2\pi) = 0$

The function attains its maximum value of $0$ at both $\pi$ and $2\pi$.

Statement (D) is correct.

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Correct statements: (A), (C) and (D) only