Let `f(x)=[x^3 - 3]`, where [.] is the greatest integer function, then the number of points in the interval (1,2) where function is discontinuous is (A) 4 (B) 5 (C) 6 (D) 7

Options

A. 4
B. 2
C. 6
D. none of these

Explanation

To determine the number of points in the interval (1, 2) where the function \( f(x) = [x^3 - 3] \) (where [.] denotes the greatest integer function) is discontinuous, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Function**:
Let \( g(x) = x^3 - 3 \). We need to analyze the behavior of \( g(x) \) in the interval \( (1, 2) \).

2. **Evaluate the Function at the Endpoints**:
- Calculate \( g(1) \):
\[ g(1) = 1^3 - 3 = 1 - 3 = -2 \]
- Calculate \( g(2) \):
\[ g(2) = 2^3 - 3 = 8 - 3 = 5 \]

3. **Determine the Range of \( g(x) \)**:
Since \( g(x) \) is a continuous and increasing function in the interval \( (1, 2) \), it will take all values between \( g(1) \) and \( g(2) \). Thus, the range of \( g(x) \) in this interval is:
\[ (-2, 5) \]

4. **Identify Integer Values in the Range**:
The integers that lie within the range \( (-2, 5) \) are:
\[ -1, 0, 1, 2, 3, 4 \]
This gives us a total of 6 integers.

5. **Identify Points of Discontinuity**:
The function \( f(x) = [g(x)] \) will be discontinuous at each integer value that \( g(x) \) takes in the interval. Since there are 6 integers in the range \( (-2, 5) \), the function \( f(x) \) will be discontinuous at these 6 points.

6. **Conclusion**:
Therefore, the number of points in the interval \( (1, 2) \) where the function \( f(x) \) is discontinuous is:
\[ \boxed{6} \]