Let `f(x)=[t a n x[cot x]],x [pi/(12),pi/(12)]` , (where [.] denotes the greatest integer less than or equal to`x` ). Then the number of points, where `f(x)` is discontinuous is a. one b. zero`` c. three d. infinite

Options

A. one
B. zero
C. three
D. infinite

Explanation

To solve the problem, we need to analyze the function \( f(x) = \lfloor \tan x \cot x \rfloor \) where \( x \) is in the interval \( \left[\frac{\pi}{12}, \frac{\pi}{2}\right] \). The notation \( \lfloor . \rfloor \) denotes the greatest integer less than or equal to the value inside.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function can be simplified:
\[ \tan x \cot x = \tan x \cdot \frac{1}{\tan x} = 1 \]
for all \( x \) where \( \tan x \) is defined and non-zero. However, we need to consider the behavior of \( \tan x \) and \( \cot x \) in the given interval.

2. **Finding the Interval**:
The interval given is \( \left[\frac{\pi}{12}, \frac{\pi}{2}\right] \). We need to check the values of \( \tan x \) and \( \cot x \) in this interval.

3. **Behavior of \( \tan x \) and \( \cot x \)**:
- At \( x = \frac{\pi}{12} \):
\[ \tan\left(\frac{\pi}{12}\right) = \frac{1}{\sqrt{3}} \quad \text{and} \quad \cot\left(\frac{\pi}{12}\right) = \sqrt{3} \]
Thus, \( \tan\left(\frac{\pi}{12}\right) \cot\left(\frac{\pi}{12}\right) = 1 \).

- As \( x \) approaches \( \frac{\pi}{2} \), \( \tan x \) approaches infinity and \( \cot x \) approaches 0.

4. **Discontinuities**:
The function \( f(x) \) will be discontinuous at points where \( \tan x \) or \( \cot x \) is undefined or where the floor function changes its value. The points of discontinuity occur when \( \tan x \cot x \) crosses integer values.

5. **Finding Critical Points**:
We need to find where \( \tan x \cot x = n \) for integers \( n \). The critical points are where \( \cot x = 0 \) (undefined) or where \( \tan x \) and \( \cot x \) yield integer results.

- \( \tan x \cot x = 1 \) is valid throughout the interval except at points where \( \tan x \) or \( \cot x \) is undefined.

6. **Calculating the Number of Discontinuities**:
We need to find the values of \( x \) in \( \left[\frac{\pi}{12}, \frac{\pi}{2}\right] \) where \( \tan x \cot x \) transitions through integers. The critical values are:
- \( \cot x = 1 \) (i.e., \( x = \frac{\pi}{4} \))
- \( \cot x = 2 \) (i.e., \( x = \cot^{-1}(2) \))
- \( \cot x = 0 \) (undefined at \( x = \frac{\pi}{2} \))

These yield three points of discontinuity.

### Conclusion:
The number of points where \( f(x) \) is discontinuous is **three**.

### Final Answer:
The correct option is **c. three**.