Understanding Increasing and Decreasing Functions

A function \( f(x) \) is considered increasing in an interval if its derivative \( f'(x) > 0 \) for all \( x \) in that interval. Conversely, a function \( f(x) \) is considered decreasing in an interval if its derivative \( f'(x) < 0 \) for all \( x \) in that interval.

For the given function \( f(x) = \sin x \), the derivative is \( f'(x) = \cos x \).

We need to analyze the sign of \( f'(x) = \cos x \) in the given intervals to determine if \( f(x) \) is increasing or decreasing.

Analysis of Statement 1: f(x) increases in the interval (0, π)

The interval is \( (0, \pi) \). We need to check the sign of \( \cos x \) in this interval.

• In the interval \( (0, \pi/2) \), which is the first quadrant, \( \cos x > 0 \). So, \( f(x) = \sin x \) is increasing in \( (0, \pi/2) \).
• In the interval \( (\pi/2, \pi) \), which is the second quadrant, \( \cos x < 0 \). So, \( f(x) = \sin x \) is decreasing in \( (\pi/2, \pi) \).

For the function \( f(x) = \sin x \) to increase in the entire interval \( (0, \pi) \), its derivative \( \cos x \) must be positive throughout the interval. Since \( \cos x \) is positive in \( (0, \pi/2) \) but negative in \( (\pi/2, \pi) \), the function does not strictly increase over the entire interval \( (0, \pi) \).

Therefore, Statement 1 is incorrect.

Analysis of Statement 2: f(x) decreases in the interval \( \left(\dfrac{5\pi}{2},3\pi\right) \)

The interval is \( \left(\dfrac{5\pi}{2},3\pi\right) \). We need to check the sign of \( \cos x \) in this interval.

• Let's analyze the angles in the interval: \( \frac{5\pi}{2} = 2\pi + \frac{\pi}{2} \) and \( 3\pi = 2\pi + \pi \).
• The interval \( \left(\dfrac{5\pi}{2},3\pi\right) \) corresponds to angles starting from just after \( \frac{\pi}{2} \) (after completing two full rotations) and ending at \( \pi \) (after completing two full rotations).
• Angles between \( \frac{5\pi}{2} \) and \( 3\pi \) are equivalent to angles between \( \frac{\pi}{2} \) and \( \pi \) in the standard unit circle representation, considering the periodicity.
• The interval \( (\pi/2, \pi) \) lies in the second quadrant.
• In the second quadrant, \( \cos x < 0 \).

Since \( \cos x < 0 \) for all \( x \) in the interval \( \left(\dfrac{5\pi}{2},3\pi\right) \), the function \( f(x) = \sin x \) is decreasing in this interval.

Therefore, Statement 2 is correct.

Conclusion from Statement Analysis

Based on the analysis of the two statements:

• Statement 1 is incorrect.
• Statement 2 is correct.

We are looking for which statement(s) are correct. Only Statement 2 is correct.

Summary of Results

Statement 1 claims \( f(x) = \sin x \) increases in \( (0, \pi) \). This is false because it increases only in \( (0, \pi/2) \) and decreases in \( (\pi/2, \pi) \).

Statement 2 claims \( f(x) = \sin x \) decreases in \( \left(\dfrac{5\pi}{2},3\pi\right) \). This is true because \( \cos x < 0 \) in this interval.

Final Answer Determination

Since only Statement 2 is correct, the option that indicates "2 only" is the correct answer.

Revision Table: Sine Function Behavior

Where \( n \) is an integer.

Additional Information: Periodicity and Derivatives

The trigonometric function \( f(x) = \sin x \) is periodic with a period of \( 2\pi \). This means its behavior repeats every \( 2\pi \) interval. Its derivative \( f'(x) = \cos x \) is also periodic with a period of \( 2\pi \).

To analyze the behavior of \( \sin x \) or \( \cos x \) over an interval that extends beyond \( [0, 2\pi) \), we can use the periodicity to map the interval back to an equivalent interval within \( [0, 2\pi) \) or \( [0, \pi) \) depending on the function's symmetry.

For \( \cos x \), \( \cos(x + 2n\pi) = \cos x \) for any integer \( n \).

In Statement 2, the interval is \( \left(\dfrac{5\pi}{2},3\pi\right) \). We can subtract \( 2\pi \) from the endpoints:

• \( \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2} \)
• \( 3\pi - 2\pi = \pi \)

So the interval \( \left(\dfrac{5\pi}{2},3\pi\right) \) behaves the same as \( (\pi/2, \pi) \) regarding the sign of \( \cos x \).

Understanding the sign of the derivative is a fundamental concept in calculus for determining where functions are increasing or decreasing, which is crucial for sketching graphs and finding local extrema.