Find the intervals in which the function given by `f(x)=sin3x, x in [0, (pi)/(2)]` is :
(a) increasing (b) decreasing.

Answer

To solve the problem of finding the intervals in which the function \( f(x) = \sin(3x) \) is increasing and decreasing on the interval \( [0, \frac{\pi}{2}] \), we will follow these steps:

### Step 1: Find the derivative of the function
We start by differentiating the function \( f(x) \):
\[ f'(x) = \frac{d}{dx}(\sin(3x)) = 3\cos(3x) \]

### Step 2: Determine where the derivative is positive or negative
To find where the function is increasing or decreasing, we need to analyze the sign of the derivative \( f'(x) \):
- The function is **increasing** where \( f'(x) > 0 \).
- The function is **decreasing** where \( f'(x) < 0 \).

### Step 3: Set the derivative to zero to find critical points
We set the derivative equal to zero to find critical points:
\[ 3\cos(3x) = 0 \]
This simplifies to:
\[ \cos(3x) = 0 \]
The cosine function is zero at odd multiples of \( \frac{\pi}{2} \):
\[ 3x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
For our interval \( [0, \frac{\pi}{2}] \), we only consider \( n = 0 \):
\[ 3x = \frac{\pi}{2} \implies x = \frac{\pi}{6} \]

### Step 4: Test intervals around the critical point
We will test the intervals \( [0, \frac{\pi}{6}) \) and \( (\frac{\pi}{6}, \frac{\pi}{2}] \) to determine the sign of \( f'(x) \).

1. **Interval \( [0, \frac{\pi}{6}) \)**:
- Choose a test point, for example, \( x = 0 \):
\[ f'(0) = 3\cos(3 \cdot 0) = 3\cos(0) = 3 > 0 \]
Thus, \( f(x) \) is increasing on \( [0, \frac{\pi}{6}) \).

2. **Interval \( (\frac{\pi}{6}, \frac{\pi}{2}] \)**:
- Choose a test point, for example, \( x = \frac{\pi}{4} \):
\[ f'\left(\frac{\pi}{4}\right) = 3\cos\left(3 \cdot \frac{\pi}{4}\right) = 3\cos\left(\frac{3\pi}{4}\right) = 3 \left(-\frac{\sqrt{2}}{2}\right) < 0 \]
Thus, \( f(x) \) is decreasing on \( (\frac{\pi}{6}, \frac{\pi}{2}] \).

### Conclusion:
- The function \( f(x) = \sin(3x) \) is **increasing** on the interval \( [0, \frac{\pi}{6}) \).
- The function \( f(x) = \sin(3x) \) is **decreasing** on the interval \( (\frac{\pi}{6}, \frac{\pi}{2}] \).