The value of a for which twice the sum of the cubes of the roots of the equation `a=(x^(2)-3)/(x-2)` attains its minimum value is (where, `a in[0, pi]`)

Options

A. greater than 4
B. less than 2
C. greater than `(7)/(4)`
D. less than 1

Answer

To solve the problem, we need to find the value of \( a \) for which twice the sum of the cubes of the roots of the equation \( a = \frac{x^2 - 3}{x - 2} \) attains its minimum value, with \( a \) in the interval \([0, \pi]\).

### Step-by-Step Solution

1. **Rewrite the Equation**:
Start with the equation given:
\[ a = \frac{x^2 - 3}{x - 2} \]
Rearranging gives:
\[ a(x - 2) = x^2 - 3 \implies x^2 - ax + (2a - 3) = 0 \]

2. **Identify the Roots**:
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). From Vieta's formulas, we know:
- Sum of the roots:
\[ \alpha + \beta = a \]
- Product of the roots:
\[ \alpha \beta = 2a - 3 \]

3. **Calculate the Sum of Cubes of the Roots**:
The sum of the cubes of the roots can be expressed using the identity:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2) \]
We can rewrite \( \alpha^2 + \beta^2 \) as:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = a^2 - 2(2a - 3) = a^2 - 4a + 6 \]
Thus,
\[ \alpha^3 + \beta^3 = a\left(a^2 - 4a + 6 - (2a - 3)\right) = a(a^2 - 6a + 9) = a(a - 3)^2 \]

4. **Twice the Sum of the Cubes**:
We need to find:
\[ 2(\alpha^3 + \beta^3) = 2a(a - 3)^2 \]

5. **Minimize the Function**:
Let \( z = 2a(a - 3)^2 \). We will find the critical points by differentiating:
\[ z = 2a(a^2 - 6a + 9) = 2a^3 - 12a^2 + 18a \]
Differentiate \( z \):
\[ \frac{dz}{da} = 6a^2 - 24a + 18 \]
Set the derivative to zero:
\[ 6a^2 - 24a + 18 = 0 \implies a^2 - 4a + 3 = 0 \]
Factor the quadratic:
\[ (a - 3)(a - 1) = 0 \]
Thus, \( a = 1 \) or \( a = 3 \).

6. **Evaluate the Function at Critical Points**:
We need to evaluate \( z \) at these points:
- For \( a = 1 \):
\[ z = 2(1)(1 - 3)^2 = 2(1)(4) = 8 \]
- For \( a = 3 \):
\[ z = 2(3)(3 - 3)^2 = 2(3)(0) = 0 \]

7. **Determine the Minimum Value**:
The minimum value of \( z \) occurs at \( a = 3 \).

### Conclusion
The value of \( a \) for which twice the sum of the cubes of the roots attains its minimum value is:
\[ \boxed{3} \]