If `a+b+c=3` and `agt0,bgt0,cgt0` then the greatest value of `a^(2)b^(3)c^(2)` is

Options

A. `(3^(4)*2^(10))/(7^(7))`
B. `(3^(10)*2^(4))/(7^(7))`
C. `(3^(2)*2^(12))/(7^(7))`
D. `(3^(12)*2^(2))/(7^(7))`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to maximize the expression \( a^2 b^3 c^2 \) under the constraint \( a + b + c = 3 \).

2. **Applying the AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative real numbers \( x_1, x_2, \ldots, x_n \):
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
We will apply this to the terms \( a, a, b, b, b, c, c \) (which gives us 7 terms).

3. **Setting up the AM-GM Inequality**:
We can express the AM-GM inequality as follows:
\[ \frac{a + a + b + b + b + c + c}{7} \geq \sqrt[7]{a^2 b^3 c^2} \]
Since \( a + b + c = 3 \), we have:
\[ \frac{2a + 3b + 2c}{7} = \frac{3}{7} \]

4. **Substituting into the Inequality**:
Thus, we can write:
\[ \frac{3}{7} \geq \sqrt[7]{a^2 b^3 c^2} \]

5. **Raising Both Sides to the Power of 7**:
Now, raising both sides to the power of 7 gives us:
\[ \left(\frac{3}{7}\right)^7 \geq a^2 b^3 c^2 \]

6. **Calculating the Maximum Value**:
Therefore, the maximum value of \( a^2 b^3 c^2 \) is:
\[ a^2 b^3 c^2 \leq \frac{3^7}{7^7} \]

7. **Finding the Exact Maximum**:
To find the exact maximum, we need to check when equality holds in the AM-GM inequality. This occurs when:
\[ a = a = b = b = b = c = c \]
Let \( a = 2x \), \( b = 3x \), \( c = 2x \). Then:
\[ 2x + 3x + 2x = 3 \implies 7x = 3 \implies x = \frac{3}{7} \]
Thus:
\[ a = \frac{6}{7}, \quad b = \frac{9}{7}, \quad c = \frac{6}{7} \]

8. **Substituting Back to Find Maximum**:
Now substituting back to find \( a^2 b^3 c^2 \):
\[ a^2 b^3 c^2 = \left(\frac{6}{7}\right)^2 \left(\frac{9}{7}\right)^3 \left(\frac{6}{7}\right)^2 \]
Simplifying this gives:
\[ = \frac{6^4 \cdot 9^3}{7^7} \]

9. **Final Calculation**:
Thus, the maximum value of \( a^2 b^3 c^2 \) is:
\[ = \frac{6^4 \cdot 9^3}{7^7} \]

### Conclusion:
The greatest value of \( a^2 b^3 c^2 \) is \( \frac{6^4 \cdot 9^3}{7^7} \).