If roots of the equation `f(x)=x^6-12 x^5+bx^4+cx^3+dx^2+ex+64=0`are positive, then
Which has the greatest absolute value ? (a) b (b) c (c) d (d) e

Options

A. b
B. c
C. d
D. e

Explanation

To solve the problem, we will analyze the polynomial given and use the properties of the roots to find the coefficients \( b, c, d, e \) and determine which has the greatest absolute value.

### Step-by-Step Solution:

1. **Identify the Polynomial**:
The polynomial is given as:
\[ f(x) = x^6 - 12x^5 + bx^4 + cx^3 + dx^2 + ex + 64 \]
We know that the roots \( x_1, x_2, x_3, x_4, x_5, x_6 \) are all positive.

2. **Sum and Product of Roots**:
From Vieta's formulas, we can find:
- The sum of the roots \( x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 12 \) (since the coefficient of \( x^5 \) is -12).
- The product of the roots \( x_1 x_2 x_3 x_4 x_5 x_6 = 64 \) (since the constant term is 64).

3. **Applying AM-GM Inequality**:
Since all roots are positive, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq \sqrt[6]{x_1 x_2 x_3 x_4 x_5 x_6} \]
Substituting the known values:
\[ \frac{12}{6} \geq \sqrt[6]{64} \]
Simplifying gives:
\[ 2 \geq 2 \]
This indicates that equality holds, which implies that all roots are equal.

4. **Finding the Roots**:
Since all roots are equal, let \( x_1 = x_2 = x_3 = x_4 = x_5 = x_6 = n \). Then:
\[ 6n = 12 \implies n = 2 \]
Thus, all roots are \( 2 \).

5. **Expanding the Polynomial**:
The polynomial can be expressed as:
\[ f(x) = (x - 2)^6 \]
Expanding this using the binomial theorem:
\[ (x - 2)^6 = x^6 - 6 \cdot 2 x^5 + 15 \cdot 4 x^4 - 20 \cdot 8 x^3 + 15 \cdot 16 x^2 - 6 \cdot 32 x + 64 \]
Simplifying the coefficients gives:
\[ = x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64 \]

6. **Identifying Coefficients**:
From the expanded polynomial, we can identify:
- \( b = 60 \)
- \( c = -160 \)
- \( d = 240 \)
- \( e = -192 \)

7. **Finding the Greatest Absolute Value**:
Now we compare the absolute values of the coefficients:
- \( |b| = 60 \)
- \( |c| = 160 \)
- \( |d| = 240 \)
- \( |e| = 192 \)

The greatest absolute value is \( |d| = 240 \).

### Conclusion:
The coefficient with the greatest absolute value is:
\[ \text{(c) } d \]