The ubiquitous AM-GM inequality has many
applications. It almost crops up in unlikely situations and
the solutions using AM-GM are truly elegant . Recall
that for n positive reals `a_(i) I = 1,2 …,`n, the AM-GM inequality tells
`(overset(n) underset(1)suma_i)/n ge ( overset(n)underset(1)proda_i)^((1)/(n))`
The special in which the inequality turns into equality
help solves many problems where at first we seem to
have not informantion to arrive at the answer .
If the equation `x^(4) - 4x^(3) + ax^(2) + bx + 1 = 0 ` has
four positive roots , then the value of `(|a|+|b|)/(a+b)` is

Options

A. (a)5
B. (b)`-5`
C. (c)`3`
D. (d)`-3`

Answer

To solve the problem, we need to analyze the polynomial equation given and apply the AM-GM inequality to find the required value of \((|a| + |b|)/(a + b)\).

### Step-by-Step Solution:

1. **Understand the Polynomial**:
The polynomial given is:
\[ x^4 - 4x^3 + ax^2 + bx + 1 = 0 \]
We need to find the values of \(a\) and \(b\) such that this equation has four positive roots.

2. **Use Vieta's Formulas**:
Let the roots of the polynomial be \(\alpha, \beta, \gamma, \delta\). By Vieta's formulas:
- The sum of the roots \(\alpha + \beta + \gamma + \delta = 4\) (coefficient of \(x^3\) with a negative sign).
- The product of the roots \(\alpha \beta \gamma \delta = 1\) (constant term).

3. **Apply AM-GM Inequality**:
By the AM-GM inequality:
\[ \frac{\alpha + \beta + \gamma + \delta}{4} \geq \sqrt[4]{\alpha \beta \gamma \delta} \]
Substituting the known values:
\[ \frac{4}{4} \geq \sqrt[4]{1} \implies 1 \geq 1 \]
This shows equality holds, which means all roots are equal:
\[ \alpha = \beta = \gamma = \delta = 1 \]

4. **Substituting Roots into the Polynomial**:
If \(\alpha = \beta = \gamma = \delta = 1\), we can substitute these values back into the polynomial:
\[ (x - 1)^4 = 0 \]
Expanding this gives:
\[ x^4 - 4x^3 + 6x^2 - 4x + 1 = 0 \]
Comparing coefficients with the original polynomial:
- Coefficient of \(x^2\) gives \(a = 6\).
- Coefficient of \(x\) gives \(b = -4\).

5. **Calculate \((|a| + |b|)/(a + b)\)**:
Now we compute:
\[ |a| = |6| = 6, \quad |b| = |-4| = 4 \]
Thus,
\[ |a| + |b| = 6 + 4 = 10 \]
And,
\[ a + b = 6 - 4 = 2 \]
Therefore,
\[ \frac{|a| + |b|}{a + b} = \frac{10}{2} = 5 \]

### Final Answer:
The value of \(\frac{|a| + |b|}{a + b}\) is \(5\).