Find the intervals in which the function `f(x) = sin x +cos x,x in [0, 2pi]` is
(i) strictly increasing, (ii) strictly decreasing.

Answer

To find the intervals in which the function \( f(x) = \sin x + \cos x \) is strictly increasing and strictly decreasing on the interval \( [0, 2\pi] \), we will follow these steps:

### Step 1: Find the derivative of the function
The first step is to differentiate the function \( f(x) \).
\[ f'(x) = \frac{d}{dx}(\sin x + \cos x) = \cos x - \sin x \]

### Step 2: Set the derivative to zero to find critical points
Next, we need to find the points where the derivative is zero, as these points will help us determine where the function changes from increasing to decreasing or vice versa.
\[ f'(x) = 0 \implies \cos x - \sin x = 0 \implies \cos x = \sin x \]
This occurs when:
\[ \tan x = 1 \implies x = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]
Within the interval \( [0, 2\pi] \), the solutions are:
\[ x = \frac{\pi}{4} \quad \text{and} \quad x = \frac{5\pi}{4} \]

### Step 3: Analyze the sign of the derivative
Now, we will analyze the sign of \( f'(x) \) in the intervals determined by the critical points \( \frac{\pi}{4} \) and \( \frac{5\pi}{4} \):
- **Interval 1:** \( [0, \frac{\pi}{4}) \)
- **Interval 2:** \( (\frac{\pi}{4}, \frac{5\pi}{4}) \)
- **Interval 3:** \( (\frac{5\pi}{4}, 2\pi] \)

**For Interval 1: \( [0, \frac{\pi}{4}) \)**
Choose a test point, for example, \( x = 0 \):
\[ f'(0) = \cos(0) - \sin(0) = 1 - 0 = 1 > 0 \]
Thus, \( f(x) \) is increasing in this interval.

**For Interval 2: \( (\frac{\pi}{4}, \frac{5\pi}{4}) \)**
Choose a test point, for example, \( x = \pi \):
\[ f'(\pi) = \cos(\pi) - \sin(\pi) = -1 - 0 = -1 < 0 \]
Thus, \( f(x) \) is decreasing in this interval.

**For Interval 3: \( (\frac{5\pi}{4}, 2\pi] \)**
Choose a test point, for example, \( x = \frac{3\pi}{2} \):
\[ f'(\frac{3\pi}{2}) = \cos(\frac{3\pi}{2}) - \sin(\frac{3\pi}{2}) = 0 - (-1) = 1 > 0 \]
Thus, \( f(x) \) is increasing in this interval.

### Step 4: Summarize the results
- The function \( f(x) \) is **strictly increasing** on the intervals:
\[ [0, \frac{\pi}{4}) \quad \text{and} \quad (\frac{5\pi}{4}, 2\pi] \]
- The function \( f(x) \) is **strictly decreasing** on the interval:
\[ (\frac{\pi}{4}, \frac{5\pi}{4}) \]

### Final Answer:
- **Strictly Increasing:** \( [0, \frac{\pi}{4}) \cup (\frac{5\pi}{4}, 2\pi] \)
- **Strictly Decreasing:** \( (\frac{\pi}{4}, \frac{5\pi}{4}) \)