If `2a+b+3c=1` and `a gt 0, b gt 0, c gt 0` , then the greatest value of `a^(4)b^(2)c^(2)"_____"`.

Answer

To find the greatest value of \( a^4 b^2 c^2 \) given the constraint \( 2a + b + 3c = 1 \) and \( a > 0, b > 0, c > 0 \), 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. **Rewrite the constraint**:
We have the equation \( 2a + b + 3c = 1 \). We can express this in a way that allows us to apply the AM-GM inequality.

2. **Express each term for AM-GM**:
We can express \( 2a \) as \( a + a \), \( b \) as \( b \), and \( 3c \) as \( c + c + c \). Thus, we rewrite the equation as:
\[ a + a + b + c + c + c = 1 \]

3. **Apply the AM-GM inequality**:
According to the AM-GM inequality, the arithmetic mean of non-negative numbers is greater than or equal to the geometric mean. Therefore, we have:
\[ \frac{a + a + b + c + c + c}{6} \geq \sqrt[6]{a^2 b c^3} \]
Substituting the left side with our constraint:
\[ \frac{1}{6} \geq \sqrt[6]{a^2 b c^3} \]

4. **Raise both sides to the power of 6**:
\[ \left(\frac{1}{6}\right)^6 \geq a^2 b c^3 \]

5. **Express \( a^4 b^2 c^2 \)**:
We want to find \( a^4 b^2 c^2 \). Notice that:
\[ a^4 b^2 c^2 = (a^2 b c^3) \cdot \frac{a^2 b}{c} \]
To maximize \( a^4 b^2 c^2 \), we need to find the maximum of \( a^2 b c^3 \).

6. **Use the constraint to find relationships**:
From the previous step, we have:
\[ a^2 b c^3 \leq \left(\frac{1}{6}\right)^6 \]
Now, we can express \( a^4 b^2 c^2 \) in terms of \( a^2 b c^3 \):
\[ a^4 b^2 c^2 = a^2 b c^3 \cdot \frac{a^2 b}{c} \]

7. **Find the maximum value**:
We can set \( a = \frac{1}{12}, b = \frac{1}{6}, c = \frac{1}{18} \) to satisfy the constraint \( 2a + b + 3c = 1 \). After substituting these values into \( a^4 b^2 c^2 \), we can find the maximum value.

### Final Calculation:
After substituting the values, we find:
\[ a^4 b^2 c^2 = \left(\frac{1}{12}\right)^4 \left(\frac{1}{6}\right)^2 \left(\frac{1}{18}\right)^2 \]

### Conclusion:
Thus, the greatest value of \( a^4 b^2 c^2 \) is achieved under the given constraints.