If the equation `x^4-4x^3+ax^2+bx+1=0`has four positive roots, then the value of (a+ b) is :

Options

A. `-4`
B. 2
C. 6
D. can not be determined

Answer

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

### Step 1: Identify the roots
Let the roots of the equation be \( \alpha, \beta, \gamma, \delta \).

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

Let this be Equation (1):
\[ \alpha + \beta + \gamma + \delta = 4 \]

### Step 3: Find the product of the roots
The product of the roots is given by:
\[ \alpha \beta \gamma \delta = \frac{\text{constant term}}{\text{coefficient of } x^4} = \frac{1}{1} = 1 \]
Let this be Equation (2):
\[ \alpha \beta \gamma \delta = 1 \]

### Step 4: Apply AM-GM inequality
Since all roots are positive, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{\alpha + \beta + \gamma + \delta}{4} \geq \sqrt[4]{\alpha \beta \gamma \delta} \]
Substituting the values from Equations (1) and (2):
\[ \frac{4}{4} \geq \sqrt[4]{1} \]
This simplifies to:
\[ 1 \geq 1 \]
Equality holds in the AM-GM inequality when all the roots are equal. Therefore, we have:
\[ \alpha = \beta = \gamma = \delta \]

### Step 5: Solve for the roots
Let \( \alpha = \beta = \gamma = \delta = k \). From Equation (1):
\[ 4k = 4 \implies k = 1 \]
Thus, all roots are equal to 1.

### Step 6: Substitute the roots back into the polynomial
Now substitute \( \alpha = 1 \) into the original equation:
\[ 1^4 - 4 \cdot 1^3 + a \cdot 1^2 + b \cdot 1 + 1 = 0 \]
This simplifies to:
\[ 1 - 4 + a + b + 1 = 0 \]
\[ a + b - 2 = 0 \]
Thus, we find:
\[ a + b = 2 \]

### Final Answer
The value of \( a + b \) is \( \boxed{2} \).