Equation `x^4+ax^3+bx^2+cx+1=0` has real roots (a,b,c are non-negative). Minimum non-negative real value of c is (a) 10 (b) 9 (c) 6 (d) 4

Options

A. 10
B. 9
C. 6
D. 4

Explanation

To find the minimum non-negative real value of \( c \) in the equation

\[ x^4 + ax^3 + bx^2 + cx + 1 = 0 \]

where \( a, b, c \) are non-negative and the equation has real roots, we can follow these steps:

### Step 1: Analyze the Roots
Given that \( a, b, c \) are non-negative, the roots of the polynomial must be negative. If any root were positive, the polynomial would not equal zero for non-negative coefficients.

Let the roots be \( x_1, x_2, x_3, x_4 \). Since these roots are negative, we can denote them as \( -y_1, -y_2, -y_3, -y_4 \) where \( y_i > 0 \).

### Step 2: Apply Vieta's Formulas
According to Vieta's formulas, we have:

- The sum of the roots \( x_1 + x_2 + x_3 + x_4 = -a \)
- The sum of the products of the roots taken two at a time \( x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 = b \)
- The sum of the products of the roots taken three at a time \( x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4 = -c \)
- The product of the roots \( x_1x_2x_3x_4 = 1 \)

### Step 3: Express \( c \) in Terms of Roots
From Vieta's, we know:

\[ -c = x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4 \]

### Step 4: Apply AM-GM Inequality
To find a lower bound for \( c \), we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality to the products of the roots taken three at a time.

The AM-GM inequality states that:

\[ \frac{x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4}{4} \geq \sqrt[4]{(x_1x_2x_3x_4)^3} \]

Since \( x_1x_2x_3x_4 = 1 \), we have:

\[ \sqrt[4]{(x_1x_2x_3x_4)^3} = \sqrt[4]{1^3} = 1 \]

Thus,

\[ \frac{-c}{4} \geq 1 \]

### Step 5: Solve for \( c \)
From the inequality, we can multiply both sides by -4 (reversing the inequality):

\[ -c \geq 4 \implies c \geq 4 \]

### Conclusion
The minimum non-negative real value of \( c \) is \( 4 \). Therefore, the answer is:

\[ \boxed{4} \]