Find the intervals in which `f(x) = sin x - cos x , "where "0 lt x lt 2 pi ` is decreasing

Options

A. `[ pi, 2 pi]`
B. `((3 pi)/(4), (7 pi)/(4))`
C. `((7 pi)/( 4),2 pi )`
D. None of these

Answer

To find the intervals in which the function \( f(x) = \sin x - \cos x \) is decreasing for \( 0 < x < 2\pi \), we will follow these steps:

### Step 1: Find the derivative of the function
To determine where the function is increasing or decreasing, we first need to find the derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}(\sin x - \cos x) = \cos x + \sin x \]

### Step 2: Set the derivative equal to zero
Next, we need to find the critical points by setting the derivative equal to zero:

\[ \cos x + \sin x = 0 \]

### Step 3: Solve for \( x \)
We can rearrange the equation:

\[ \sin x = -\cos x \]

Dividing both sides by \( \cos x \) (where \( \cos x \neq 0 \)), we get:

\[ \tan x = -1 \]

### Step 4: Find the solutions for \( x \)
The general solutions for \( \tan x = -1 \) are:

\[ x = \frac{3\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z} \]

Within the interval \( 0 < x < 2\pi \), the specific solutions are:

\[ x = \frac{3\pi}{4} \quad \text{and} \quad x = \frac{7\pi}{4} \]

### Step 5: Test intervals around the critical points
Now we will test the intervals determined by the critical points \( \frac{3\pi}{4} \) and \( \frac{7\pi}{4} \):

1. **Interval \( (0, \frac{3\pi}{4}) \)**:
- Choose \( x = 0 \):
\[ f'(0) = \cos(0) + \sin(0) = 1 + 0 = 1 \quad (\text{positive, increasing}) \]

2. **Interval \( (\frac{3\pi}{4}, \frac{7\pi}{4}) \)**:
- Choose \( x = \pi \):
\[ f'(\pi) = \cos(\pi) + \sin(\pi) = -1 + 0 = -1 \quad (\text{negative, decreasing}) \]

3. **Interval \( (\frac{7\pi}{4}, 2\pi) \)**:
- Choose \( x = \frac{3\pi}{2} \):
\[ f'\left(\frac{3\pi}{2}\right) = \cos\left(\frac{3\pi}{2}\right) + \sin\left(\frac{3\pi}{2}\right) = 0 - 1 = -1 \quad (\text{negative, decreasing}) \]

### Step 6: Conclusion
From our analysis, we find that the function \( f(x) \) is decreasing in the intervals:

\[ \left(\frac{3\pi}{4}, \frac{7\pi}{4}\right) \quad \text{and} \quad \left(\frac{7\pi}{4}, 2\pi\right) \]

Thus, the final answer is:

\[ f(x) \text{ is decreasing in } \left(\frac{3\pi}{4}, \frac{7\pi}{4}\right) \text{ and } \left(\frac{7\pi}{4}, 2\pi\right) \]