If the equation `x^(3)-3ax^(2)+3bx-c=0` has positive and distinct roots, then

Options

A. `a^(2)gt b`
B. `ab gt c^(2)`
C. `a^(3)gt c^(2)`
D. `a^(3)gt b^(2)gt c`

Answer

To solve the problem, we need to analyze the cubic equation given by \( x^3 - 3ax^2 + 3bx - c = 0 \) and find the conditions under which it has positive and distinct roots. We will use Vieta's formulas and some inequalities to derive the necessary conditions.

### Step-by-Step Solution:

1. **Identify the Roots:**
Let the roots of the cubic equation be \( \alpha, \beta, \gamma \). According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta + \gamma = 3a \)
- The sum of the product of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = 3b \)
- The product of the roots \( \alpha\beta\gamma = c \)

2. **Apply the AM-GM Inequality:**
Since the roots are positive, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{\alpha + \beta + \gamma}{3} \geq \sqrt[3]{\alpha\beta\gamma} \]
Substituting the values from Vieta's:
\[ \frac{3a}{3} \geq \sqrt[3]{c} \implies a \geq \sqrt[3]{c} \]

3. **Use the Condition for Distinct Roots:**
For the cubic equation to have distinct roots, we need to ensure that the discriminant is positive. However, we can also derive a relationship using the squares of the sums:
\[ (\alpha + \beta + \gamma)^2 = \alpha^2 + \beta^2 + \gamma^2 + 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
Rearranging gives:
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
Substituting the values:
\[ \alpha^2 + \beta^2 + \gamma^2 = (3a)^2 - 2(3b) = 9a^2 - 6b \]

4. **Ensure Positivity:**
For the roots to be distinct, we need:
\[ \alpha^2 + \beta^2 + \gamma^2 - (\alpha\beta + \beta\gamma + \gamma\alpha) > 0 \]
This simplifies to:
\[ 9a^2 - 6b - 3b > 0 \implies 9a^2 - 9b > 0 \implies a^2 > b \]

5. **Final Conclusion:**
From the analysis, we conclude that for the cubic equation \( x^3 - 3ax^2 + 3bx - c = 0 \) to have positive and distinct roots, the following condition must hold:
\[ a^2 > b \]