The number of points where `f(x) = [sin x + cosx]` (where [.] denotes the greatest integer function) `x in (0,2pi)` is not continuous is

Options

A. (A) 3
B. (B) 4
C. (C) 5
D. (D) 6

Answer

To determine the number of points where the function \( f(x) = [\sin x + \cos x] \) (where \([.]\) denotes the greatest integer function) is not continuous in the interval \( (0, 2\pi) \), we can follow these steps:

### Step 1: Analyze the function \( \sin x + \cos x \)

The first step is to rewrite the expression \( \sin x + \cos x \) in a more manageable form. We can use the identity:

\[ \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]

This transformation helps us understand the range of the function.

### Step 2: Determine the range of \( \sin x + \cos x \)

The maximum value of \( \sin x + \cos x \) occurs when \( \sin\left(x + \frac{\pi}{4}\right) = 1 \), which gives us:

\[ \max(\sin x + \cos x) = \sqrt{2} \]

The minimum value occurs when \( \sin\left(x + \frac{\pi}{4}\right) = -1 \), giving us:

\[ \min(\sin x + \cos x) = -\sqrt{2} \]

Thus, the range of \( \sin x + \cos x \) is \( [-\sqrt{2}, \sqrt{2}] \).

### Step 3: Identify the integer values in the range

The greatest integer function \([y]\) is not continuous at integer values of \(y\). Therefore, we need to identify the integer values that fall within the range of \( \sin x + \cos x \):

The integer values in the range \( [-\sqrt{2}, \sqrt{2}] \) are \( -1, 0, 1 \).

### Step 4: Find points of discontinuity

The function \( f(x) = [\sin x + \cos x] \) will be discontinuous at points where \( \sin x + \cos x \) crosses these integer values. We need to solve the following equations:

1. \( \sin x + \cos x = -1 \)
2. \( \sin x + \cos x = 0 \)
3. \( \sin x + \cos x = 1 \)

### Step 5: Solve each equation

1. **For \( \sin x + \cos x = -1 \)**:
- This occurs when \( \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) = -1 \).
- Thus, \( \sin\left(x + \frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \).
- This gives us solutions in the interval \( (0, 2\pi) \).

2. **For \( \sin x + \cos x = 0 \)**:
- This occurs when \( \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) = 0 \).
- Thus, \( \sin\left(x + \frac{\pi}{4}\right) = 0 \).
- This gives us solutions in the interval \( (0, 2\pi) \).

3. **For \( \sin x + \cos x = 1 \)**:
- This occurs when \( \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) = 1 \).
- Thus, \( \sin\left(x + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).
- This gives us solutions in the interval \( (0, 2\pi) \).

### Step 6: Count the points of discontinuity

After solving these equations, we find that there are 5 points in total where \( f(x) \) is not continuous in the interval \( (0, 2\pi) \).

### Conclusion

The number of points where \( f(x) = [\sin x + \cos x] \) is not continuous in the interval \( (0, 2\pi) \) is **5**.

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