Number of points of discontinuity of `f(x) =[x^(3)+3x^(2)+7x+2],` where [.] represents the greatest integer function in `[0,1]` is ___________.

Answer

To find the number of points of discontinuity of the function \( f(x) = \lfloor x^3 + 3x^2 + 7x + 2 \rfloor \) in the interval \([0, 1]\), where \(\lfloor . \rfloor\) denotes the greatest integer function, we can follow these steps:

### Step 1: Define the function
The function we are analyzing is:
\[ f(x) = \lfloor g(x) \rfloor \]
where \( g(x) = x^3 + 3x^2 + 7x + 2 \).

### Step 2: Find the values of \( g(x) \) at the endpoints of the interval
We need to evaluate \( g(x) \) at the endpoints \( x = 0 \) and \( x = 1 \):
- For \( x = 0 \):
\[ g(0) = 0^3 + 3(0^2) + 7(0) + 2 = 2 \]
- For \( x = 1 \):
\[ g(1) = 1^3 + 3(1^2) + 7(1) + 2 = 1 + 3 + 7 + 2 = 13 \]

### Step 3: Determine the range of \( g(x) \) in \([0, 1]\)
Since \( g(x) \) is a continuous function and we have calculated:
\[ g(0) = 2 \quad \text{and} \quad g(1) = 13 \]
The function \( g(x) \) will take all values between 2 and 13 as \( x \) varies from 0 to 1.

### Step 4: Identify integer values in the range
The integer values that \( g(x) \) can take between 2 and 13 are:
\[ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 \]
This gives us a total of \( 13 - 2 + 1 = 12 \) integer values.

### Step 5: Points of discontinuity
The greatest integer function \( \lfloor g(x) \rfloor \) is discontinuous at every integer value that \( g(x) \) takes. Hence, since \( g(x) \) takes on 12 integer values, \( f(x) \) will be discontinuous at these 12 points.

### Conclusion
The number of points of discontinuity of \( f(x) \) in the interval \([0, 1]\) is:
\[ \boxed{12} \]