The points of discontinuities of `f (x)= [(6x)/(pi)]cos [(3x)/(pi)]"in" [(pi)/(6), pi]` is/are:(where [.] denotes greattest integer function)

Options

A. `pi/6`
B. `pi/3`
C. `pi/2`
D. `pi`

Answer

To find the points of discontinuity of the function \( f(x) = \left\lfloor \frac{6x}{\pi} \right\rfloor \cos\left(\left\lfloor \frac{3x}{\pi} \right\rfloor\right) \) in the interval \(\left[\frac{\pi}{6}, \pi\right]\), we will analyze the components of the function.

### Step 1: Identify the components of the function
The function consists of two parts:
1. The greatest integer function \( \left\lfloor \frac{6x}{\pi} \right\rfloor \)
2. The cosine function multiplied by the greatest integer function \( \left\lfloor \frac{3x}{\pi} \right\rfloor \)

### Step 2: Determine where \( \left\lfloor \frac{6x}{\pi} \right\rfloor \) is discontinuous
The greatest integer function \( \left\lfloor x \right\rfloor \) is discontinuous at integer values. Therefore, we need to find the values of \( x \) for which \( \frac{6x}{\pi} \) is an integer.

Let \( z = \frac{6x}{\pi} \), then:
\[ x = \frac{\pi z}{6} \]
We need to find integer values of \( z \) such that \( x \) lies in the interval \(\left[\frac{\pi}{6}, \pi\right]\).

### Step 3: Calculate the bounds for \( z \)
1. For \( x = \frac{\pi}{6} \):
\[ z = \frac{6 \cdot \frac{\pi}{6}}{\pi} = 1 \]
2. For \( x = \pi \):
\[ z = \frac{6 \cdot \pi}{\pi} = 6 \]

Thus, \( z \) can take integer values \( 1, 2, 3, 4, 5, 6 \).

### Step 4: Calculate corresponding \( x \) values
Now, we calculate \( x \) for each integer \( z \):
- For \( z = 1 \):
\[ x = \frac{\pi \cdot 1}{6} = \frac{\pi}{6} \]
- For \( z = 2 \):
\[ x = \frac{\pi \cdot 2}{6} = \frac{\pi}{3} \]
- For \( z = 3 \):
\[ x = \frac{\pi \cdot 3}{6} = \frac{\pi}{2} \]
- For \( z = 4 \):
\[ x = \frac{\pi \cdot 4}{6} = \frac{2\pi}{3} \]
- For \( z = 5 \):
\[ x = \frac{\pi \cdot 5}{6} = \frac{5\pi}{6} \]
- For \( z = 6 \):
\[ x = \frac{\pi \cdot 6}{6} = \pi \]

### Step 5: List the points of discontinuity
The points of discontinuity of \( f(x) \) in the interval \(\left[\frac{\pi}{6}, \pi\right]\) are:
- \( \frac{\pi}{6} \)
- \( \frac{\pi}{3} \)
- \( \frac{\pi}{2} \)
- \( \frac{2\pi}{3} \)
- \( \frac{5\pi}{6} \)
- \( \pi \)

### Final Answer
The points of discontinuity of \( f(x) \) in the interval \(\left[\frac{\pi}{6}, \pi\right]\) are:
\[ \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{5\pi}{6}, \pi \]