if the equation `x^(4)-4x^(3)+ax^(2)+bx+1=0` has four positive roots, then find a.

Explanation

To solve the equation \(x^4 - 4x^3 + ax^2 + bx + 1 = 0\) under the condition that it has four positive roots, we can follow these steps:

### Step 1: Identify the roots
Let the four positive roots be \( \alpha, \beta, \gamma, \lambda \).

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta + \gamma + \lambda = -\frac{\text{coefficient of } x^3}{\text{coefficient of } x^4} = -\frac{-4}{1} = 4 \).
- The product of the roots \( \alpha \beta \gamma \lambda = \frac{\text{constant term}}{\text{coefficient of } x^4} = \frac{1}{1} = 1 \).

### Step 3: Apply the AM-GM inequality
Since all roots are positive, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{\alpha + \beta + \gamma + \lambda}{4} \geq \sqrt[4]{\alpha \beta \gamma \lambda} \]
Substituting the known values:
\[ \frac{4}{4} \geq \sqrt[4]{1} \]
This simplifies to:
\[ 1 \geq 1 \]
This equality holds when all roots are equal, i.e., \( \alpha = \beta = \gamma = \lambda \).

### Step 4: Set the roots equal
Let \( \alpha = \beta = \gamma = \lambda = k \). Then:
\[ 4k = 4 \implies k = 1 \]
Thus, all roots are \( 1 \).

### Step 5: Form the polynomial
The polynomial can be expressed as:
\[ (x - 1)^4 = 0 \]
Expanding this, we have:
\[ (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 \]

### Step 6: Compare coefficients
Now, we compare this with the original polynomial:
\[ x^4 - 4x^3 + ax^2 + bx + 1 \]
From the expansion, we see:
- The coefficient of \(x^2\) gives us \(a = 6\).
- The coefficient of \(x\) gives us \(b = -4\).

### Final Answer
Thus, the value of \(a\) is:
\[ \boxed{6} \]