Let `f(x) =(x^(2)-3x+ 2) (x ^(2) + 3x +2) and alpha, beta, gamma` satisfy `alpha lt beta gamma` are the roots of `f '(x)=0` then which of the following is/are correct ([.] denots greatest integer function) ?

Options

A. `[alpha]=-2`
B. `[beta]=-1`
C. `[beta]=0`
D. `[alpha]=1`

Answer

To solve the problem, we need to find the roots of the derivative of the function \( f(x) = (x^2 - 3x + 2)(x^2 + 3x + 2) \) and then determine the greatest integer function values for those roots.

### Step 1: Differentiate \( f(x) \)

Using the product rule, we differentiate \( f(x) \):

\[ f'(x) = (x^2 - 3x + 2)'(x^2 + 3x + 2) + (x^2 - 3x + 2)(x^2 + 3x + 2)' \]

Calculating the derivatives:

1. \( (x^2 - 3x + 2)' = 2x - 3 \)
2. \( (x^2 + 3x + 2)' = 2x + 3 \)

Substituting these into the product rule gives:

\[ f'(x) = (2x - 3)(x^2 + 3x + 2) + (x^2 - 3x + 2)(2x + 3) \]

### Step 2: Expand \( f'(x) \)

Now, we will expand both terms:

1. Expanding \( (2x - 3)(x^2 + 3x + 2) \):

\[ = 2x^3 + 6x^2 + 4x - 3x^2 - 9x - 6 = 2x^3 + 3x^2 - 5x - 6 \]

2. Expanding \( (x^2 - 3x + 2)(2x + 3) \):

\[ = 2x^3 + 3x^2 - 6x - 9 + 2 = 2x^3 + 3x^2 - 6x - 9 \]

Combining both expansions:

\[ f'(x) = (2x^3 + 3x^2 - 5x - 6) + (2x^3 + 3x^2 - 6x - 9) \]

This simplifies to:

\[ f'(x) = 4x^3 - 10x \]

### Step 3: Set \( f'(x) = 0 \)

Setting the derivative equal to zero:

\[ 4x^3 - 10x = 0 \]

Factoring out \( 2x \):

\[ 2x(2x^2 - 5) = 0 \]

This gives us:

\[ x = 0 \quad \text{or} \quad 2x^2 - 5 = 0 \]

Solving \( 2x^2 - 5 = 0 \):

\[ 2x^2 = 5 \quad \Rightarrow \quad x^2 = \frac{5}{2} \quad \Rightarrow \quad x = \pm \sqrt{\frac{5}{2}} = \pm \frac{\sqrt{10}}{2} \]

### Step 4: Identify the roots

Thus, the roots of \( f'(x) = 0 \) are:

\[ \alpha = -\frac{\sqrt{10}}{2}, \quad \beta = 0, \quad \gamma = \frac{\sqrt{10}}{2} \]

### Step 5: Calculate the greatest integer function values

1. **For \( \alpha \)**:
- \( \alpha \approx -1.5811 \)
- \( \lfloor \alpha \rfloor = -2 \)

2. **For \( \beta \)**:
- \( \beta = 0 \)
- \( \lfloor \beta \rfloor = 0 \)

3. **For \( \gamma \)**:
- \( \gamma \approx 1.5811 \)
- \( \lfloor \gamma \rfloor = 1 \)

### Conclusion

The greatest integer function values are:

- \( \lfloor \alpha \rfloor = -2 \)
- \( \lfloor \beta \rfloor = 0 \)
- \( \lfloor \gamma \rfloor = 1 \)

### Final Answer

- Option A: \( \lfloor \alpha \rfloor = -2 \) (Correct)
- Option B: \( \lfloor \beta \rfloor = 0 \) (Correct)
- Option C: \( \lfloor \gamma \rfloor = 1 \) (Correct)