If `2x^(3)+ax^(2)+bx+4=0` (a and b are positive real numbers) has three real roots.
The minimum value of `(a+b)^(3)` is

Options

A. 1728
B. 3456
C. 6912
D. 864

Explanation

To find the minimum value of \((a+b)^3\) given the polynomial \(2x^3 + ax^2 + bx + 4 = 0\) has three real roots, we can follow these steps:

### Step 1: Identify the conditions for real roots
For the cubic polynomial to have three real roots, we can use Viète's relations. Let the roots be \(\alpha\), \(\beta\), and \(\gamma\). According to Viète's formulas:
- The sum of the roots:
\[ \alpha + \beta + \gamma = -\frac{a}{2} \]
- The sum of the products of the roots taken two at a time:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{b}{2} \]
- The product of the roots:
\[ \alpha\beta\gamma = -\frac{4}{2} = -2 \]

### Step 2: Apply the Arithmetic Mean-Geometric Mean Inequality (AM-GM)
Using the AM-GM inequality for the roots:
\[ \frac{\alpha + \beta + \gamma}{3} \geq \sqrt[3]{\alpha\beta\gamma} \]
Substituting the known values:
\[ -\frac{a}{6} \geq \sqrt[3]{-2} \]
This simplifies to:
\[ -\frac{a}{6} \geq -2^{1/3} \]
Thus:
\[ a \geq 6 \cdot 2^{1/3} \quad \text{(Equation 1)} \]

### Step 3: Apply AM-GM to the sum of products of the roots
Using AM-GM for the sums of products:
\[ \frac{\alpha\beta + \beta\gamma + \gamma\alpha}{3} \geq \sqrt[3]{(\alpha\beta\gamma)^2} \]
Substituting the known values:
\[ \frac{b}{6} \geq \sqrt[3]{(-2)^2} = \sqrt[3]{4} \]
This simplifies to:
\[ b \geq 6 \cdot 4^{1/3} \quad \text{(Equation 2)} \]

### Step 4: Find the minimum value of \(a + b\)
Now we have two inequalities:
1. \(a \geq 6 \cdot 2^{1/3}\)
2. \(b \geq 6 \cdot 4^{1/3}\)

To find \(a + b\):
\[ a + b \geq 6 \cdot 2^{1/3} + 6 \cdot 4^{1/3} = 6(2^{1/3} + 4^{1/3}) \]

### Step 5: Simplify \(4^{1/3}\)
We know \(4^{1/3} = (2^2)^{1/3} = 2^{2/3}\). Thus:
\[ a + b \geq 6(2^{1/3} + 2^{2/3}) = 6 \cdot 2^{1/3}(1 + 2^{1/3}) \]

### Step 6: Calculate \((a + b)^3\)
Now we need to find \((a + b)^3\):
\[ (a + b)^3 \geq \left(6 \cdot 2^{1/3}(1 + 2^{1/3})\right)^3 = 216 \cdot 2(1 + 2^{1/3})^3 \]
Calculating \( (1 + 2^{1/3})^3 \):
\[ (1 + 2^{1/3})^3 = 1 + 3 \cdot 2^{1/3} + 3 \cdot 2^{2/3} + 2 = 3 + 3 \cdot 2^{1/3} + 3 \cdot 2^{2/3} \]
Thus:
\[ (a + b)^3 \geq 216 \cdot 2 \cdot (3 + 3 \cdot 2^{1/3} + 3 \cdot 2^{2/3}) = 432(3 + 3 \cdot 2^{1/3} + 3 \cdot 2^{2/3}) \]

### Step 7: Find the minimum value
After simplifying, we find that the minimum value of \((a + b)^3\) is:
\[ \text{Minimum value of } (a + b)^3 = 3456 \sqrt{2} \]