Let `f(x)=[2x^3-6]`, where [x] is the greatest integer less than or equal to x. Then the number of points in (1,2) where f is discontinuous is

Options

A. 5
B. 7
C. 13
D. 12

Answer

To solve the problem, we need to analyze the function \( f(x) = \lfloor 2x^3 - 6 \rfloor \) where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). We want to determine the number of points in the open interval \( (1, 2) \) where this function is discontinuous.

### Step-by-Step Solution:

1. **Identify the function**:
The function is given as \( f(x) = \lfloor 2x^3 - 6 \rfloor \).

2. **Determine the values of \( f(x) \) at the endpoints of the interval**:
- For \( x = 1 \):
\[ f(1) = \lfloor 2(1)^3 - 6 \rfloor = \lfloor 2 - 6 \rfloor = \lfloor -4 \rfloor = -4 \]
- For \( x = 2 \):
\[ f(2) = \lfloor 2(2)^3 - 6 \rfloor = \lfloor 2 \cdot 8 - 6 \rfloor = \lfloor 16 - 6 \rfloor = \lfloor 10 \rfloor = 10 \]

3. **Determine the range of \( f(x) \) in the interval \( (1, 2) \)**:
- As \( x \) varies from 1 to 2, \( 2x^3 \) varies from \( 2 \) to \( 16 \).
- Therefore, \( 2x^3 - 6 \) varies from \( -4 \) to \( 10 \).

4. **Identify the integer values in the range**:
- The integer values between \( -4 \) and \( 10 \) are:
\[ -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \]
- This gives us the integers: \( -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \) (note that we do not include \( 10 \) since the interval is open).

5. **Count the number of integers**:
- The integers from \( -4 \) to \( 9 \) are \( -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \).
- There are a total of \( 14 \) integers.

6. **Determine points of discontinuity**:
- The function \( f(x) \) is discontinuous at each integer value within the range, except for the endpoints \( -4 \) and \( 10 \) since they are not included in the open interval \( (1, 2) \).
- Thus, the points of discontinuity are at \( -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \).

7. **Conclusion**:
- The total number of points in the interval \( (1, 2) \) where \( f \) is discontinuous is \( 13 \).

### Final Answer:
The number of points in \( (1, 2) \) where \( f \) is discontinuous is \( 13 \).