If the roots of the equaion `x^4-12 x^3+c x^2+dx+81=0` are positive then the value of c is The value of d is. Roots of the equation 2cx+d=0 is

Options

A. `-27`
B. `27`
C. `-54`
D. `54`

Explanation

To solve the problem step by step, we will analyze the polynomial equation given and use Vieta's formulas to find the values of \( c \) and \( d \), and then determine the roots of the equation \( 2cx + d = 0 \).

### Step 1: Identify the polynomial and its roots
The polynomial given is:
\[ x^4 - 12x^3 + cx^2 + dx + 81 = 0 \]
Let the roots of this polynomial be \( x_1, x_2, x_3, x_4 \).

### Step 2: Use Vieta's formulas
From Vieta's formulas, we know:
1. The sum of the roots:
\[ x_1 + x_2 + x_3 + x_4 = 12 \]
2. The sum of the products of the roots taken two at a time:
\[ x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 = c \]
3. The sum of the products of the roots taken three at a time:
\[ x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4 = -d \]
4. The product of the roots:
\[ x_1x_2x_3x_4 = 81 \]

### Step 3: Apply the Arithmetic Mean-Geometric Mean Inequality
Since all roots are positive, we can apply the AM-GM inequality:
\[ \frac{x_1 + x_2 + x_3 + x_4}{4} \geq \sqrt[4]{x_1x_2x_3x_4} \]
Substituting the known values:
\[ \frac{12}{4} = 3 \quad \text{and} \quad \sqrt[4]{81} = 3 \]
Since equality holds, we conclude that:
\[ x_1 = x_2 = x_3 = x_4 = 3 \]

### Step 4: Calculate \( c \)
Now we can calculate \( c \):
\[ c = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 \]
Substituting \( x_1 = x_2 = x_3 = x_4 = 3 \):
\[ c = 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 = 6 \cdot 9 = 54 \]

### Step 5: Calculate \( d \)
Next, we calculate \( d \):
\[ -d = x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4 \]
Substituting \( x_1 = x_2 = x_3 = x_4 = 3 \):
\[ -d = 3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3 = 4 \cdot 27 = 108 \]
Thus, \( d = -108 \).

### Step 6: Solve for the roots of \( 2cx + d = 0 \)
Now we need to find the roots of the equation:
\[ 2cx + d = 0 \]
Substituting \( c = 54 \) and \( d = -108 \):
\[ 2 \cdot 54x - 108 = 0 \]
This simplifies to:
\[ 108x - 108 = 0 \implies 108x = 108 \implies x = 1 \]

### Final Answers
- The value of \( c \) is \( 54 \).
- The value of \( d \) is \( -108 \).
- The root of the equation \( 2cx + d = 0 \) is \( x = 1 \).