If a, b, c are positive real number such that `lamba` abc is the minimum value of `a(b^(2)+c^(2))+b(c^(2)+a^(2))+c(a^(2)+b^(2))`, then `lambda`=

Options

A. 1
B. 2
C. 3
D. 6

Explanation

To find the minimum value of the expression \( a(b^2 + c^2) + b(c^2 + a^2) + c(a^2 + b^2) \) and express it in the form \( \lambda abc \), we will use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Identify the Expression**:
We need to minimize the expression:
\[ S = a(b^2 + c^2) + b(c^2 + a^2) + c(a^2 + b^2) \]

2. **Rearranging the Expression**:
We can rewrite \( S \) as:
\[ S = ab^2 + ac^2 + bc^2 + ba^2 + ca^2 + cb^2 \]

3. **Applying AM-GM Inequality**:
We can apply the AM-GM inequality on the three pairs:
- \( ab^2 + ac^2 \)
- \( bc^2 + ba^2 \)
- \( ca^2 + cb^2 \)

According to AM-GM, we have:
\[ \frac{ab^2 + ac^2}{2} \geq \sqrt{ab^2 \cdot ac^2} = a \cdot b \cdot c \cdot \sqrt{bc} \]
Similarly, we can apply AM-GM to the other pairs.

4. **Combining Results**:
By applying AM-GM to all pairs, we get:
\[ S \geq 3 \sqrt[3]{(ab^2)(ac^2)(bc^2)(ba^2)(ca^2)(cb^2)} \]

The product simplifies to:
\[ (abc)^6 \]
Hence, we can express the inequality as:
\[ S \geq 6abc \]

5. **Finding the Minimum Value**:
The minimum value of \( S \) is thus \( 6abc \).

6. **Identifying \( \lambda \)**:
We are given that this minimum value can be expressed as \( \lambda abc \). By comparing:
\[ \lambda abc = 6abc \]
We find that:
\[ \lambda = 6 \]

### Conclusion:
The value of \( \lambda \) is:
\[ \boxed{6} \]