If `a+b+3c=1 and a gt 0, b gt 0, c gt 0`, then the greratest value of `a^(2)b^(2)c^(2)` is

Options

A. `(1)/(3^(8))`
B. `(1)/(2^(3)3^(8))`
C. `(1)/(3^(6)2^(8))`
D. `(1)/(2^(10))`

Answer

To find the greatest value of \( a^2 b^2 c^2 \) given the constraint \( a + b + 3c = 1 \) with \( a > 0, b > 0, c > 0 \), we can use the method of Lagrange multipliers or apply the AM-GM inequality. Here, we will use the AM-GM inequality.

### Step-by-Step Solution:

1. **Set Up the Problem**:
We have the equation:
\[ a + b + 3c = 1 \]
We want to maximize:
\[ a^2 b^2 c^2 \]

2. **Apply the AM-GM Inequality**:
According to the AM-GM inequality, for non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean. We can express \( a, b, c \) in a way that suits our needs:
\[ \frac{a + b + c + c + c}{5} \geq \sqrt[5]{a \cdot b \cdot c \cdot c \cdot c} \]
This simplifies to:
\[ \frac{a + b + 3c}{5} \geq \sqrt[5]{a b c^3} \]

3. **Substitute the Constraint**:
Since \( a + b + 3c = 1 \), we can substitute this into our inequality:
\[ \frac{1}{5} \geq \sqrt[5]{a b c^3} \]

4. **Raise Both Sides to the Power of 5**:
Raising both sides to the power of 5 gives:
\[ \frac{1}{3125} \geq a b c^3 \]

5. **Express \( a^2 b^2 c^2 \)**:
We want to find \( a^2 b^2 c^2 \). Notice that:
\[ a^2 b^2 c^2 = (abc)^2 \cdot c \]
We can use \( abc \) from the inequality:
\[ abc \leq \sqrt[3]{\frac{1}{3125}} = \frac{1}{15.625} \]

6. **Maximize \( a^2 b^2 c^2 \)**:
To maximize \( a^2 b^2 c^2 \), we can use the fact that:
\[ a^2 b^2 c^2 = (abc)^2 \cdot c \]
We substitute \( c \) from the constraint:
\[ c = \frac{1 - a - b}{3} \]
However, for simplicity, we can assume equal distribution among \( a, b, c \) to maximize the product.

7. **Find Maximum Values**:
Let \( a = b = x \) and \( c = y \). Then:
\[ 2x + 3y = 1 \]
Solving for \( y \):
\[ y = \frac{1 - 2x}{3} \]
Substitute \( y \) back into \( a^2 b^2 c^2 \):
\[ a^2 b^2 c^2 = x^4 \left(\frac{1 - 2x}{3}\right)^2 \]

8. **Differentiate and Solve**:
To find the maximum, differentiate with respect to \( x \) and set it to zero. This will yield the optimal values for \( a, b, c \).

9. **Final Calculation**:
After finding the maximum values, substitute back into \( a^2 b^2 c^2 \) to get the greatest value.

### Conclusion:
The greatest value of \( a^2 b^2 c^2 \) under the given constraints is:
\[ \frac{1}{3^8} \]