The equation `x^(3) - 3x + [a] = 0` , will have three real and distinct roots if
(where [*] denotes the greatest integer function)

Options

A. `a in (-oo,2)`
B. `a in (0,2)`
C. `a in (-oo , 2) cup (0 , oo)`
D. `a in [-1,2)`

Answer

To determine the conditions under which the equation \( x^3 - 3x + [a] = 0 \) has three real and distinct roots, we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x^3 - 3x + [a] \).

### Step 2: Find the derivative
To find the critical points, we first differentiate \( f(x) \):
\[ f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x - 1)(x + 1) \]
Setting the derivative to zero gives us the critical points:
\[ f'(x) = 0 \implies x = -1, 1 \]

### Step 3: Evaluate the function at critical points
Next, we evaluate \( f(x) \) at the critical points \( x = -1 \) and \( x = 1 \):
\[ f(-1) = (-1)^3 - 3(-1) + [a] = -1 + 3 + [a] = 2 + [a] \]
\[ f(1) = (1)^3 - 3(1) + [a] = 1 - 3 + [a] = -2 + [a] \]

### Step 4: Set conditions for three distinct roots
For \( f(x) \) to have three real and distinct roots, the following conditions must hold:
1. \( f(-1) > 0 \) (which ensures that the function is positive at \( x = -1 \))
2. \( f(1) < 0 \) (which ensures that the function is negative at \( x = 1 \))

### Step 5: Solve the inequalities
From the first condition:
\[ 2 + [a] > 0 \implies [a] > -2 \]
This implies:
\[ [a] \geq -1 \quad \text{(since [a] is an integer)} \]

From the second condition:
\[ -2 + [a] < 0 \implies [a] < 2 \]
This implies:
\[ [a] \leq 1 \quad \text{(since [a] is an integer)} \]

### Step 6: Combine the conditions
Combining the two conditions, we have:
\[ -1 \leq [a] < 2 \]
Thus, the values of \( [a] \) can be \( -1, 0, 1 \).

### Final Answer
The equation \( x^3 - 3x + [a] = 0 \) will have three real and distinct roots if:
\[ [a] \in \{-1, 0, 1\} \]