if `alpha` is a real root of `2x^3-3x^2 + 6x + 6 = 0,` then find `[alpha`]` where [] denotes the greatest integer function.

Options

A. 0
B. -1
C. 1
D. -2

Explanation

To find the greatest integer function of the real root \( \alpha \) of the equation \( 2x^3 - 3x^2 + 6x + 6 = 0 \), we will follow these steps:

### Step 1: Define the function
Let \( f(x) = 2x^3 - 3x^2 + 6x + 6 \).

### Step 2: Find the derivative
To analyze the behavior of the function, we need to find its derivative:
\[ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 + 6x + 6) = 6x^2 - 6x + 6 \]

### Step 3: Analyze the derivative
Next, we will check if the derivative has any real roots by calculating its discriminant:
\[ D = b^2 - 4ac = (-6)^2 - 4 \cdot 6 \cdot 6 = 36 - 144 = -108 \]
Since the discriminant \( D < 0 \), the quadratic \( 6x^2 - 6x + 6 \) has no real roots, which means \( f'(x) \) does not change sign and is always positive. Therefore, \( f(x) \) is a strictly increasing function.

### Step 4: Evaluate the function at specific points
Now we will evaluate \( f(x) \) at some points to locate the root:
- Calculate \( f(0) \):
\[ f(0) = 2(0)^3 - 3(0)^2 + 6(0) + 6 = 6 \]
- Calculate \( f(-1) \):
\[ f(-1) = 2(-1)^3 - 3(-1)^2 + 6(-1) + 6 = -2 - 3 - 6 + 6 = -5 \]

### Step 5: Determine the interval for the root
Since \( f(-1) = -5 \) and \( f(0) = 6 \), and \( f(x) \) is strictly increasing, there is a root \( \alpha \) in the interval \( (-1, 0) \).

### Step 6: Find the greatest integer function
Since \( \alpha \) lies between -1 and 0, the greatest integer less than or equal to \( \alpha \) is:
\[ [\alpha] = -1 \]

### Final Answer
Thus, the greatest integer function of \( \alpha \) is:
\[ \boxed{-1} \]