If `alpha+beta, gamma epsilon R^(+)` such that `alpha gamma =1/(beta)` then maximum value of `alpha^(beta+gamma), .beta^(beta+gamma).gamma^(alpha+beta)` is_____

Explanation

To solve the problem, we need to find the maximum value of the expression \( \alpha^{\beta + \gamma} \cdot \beta^{\beta + \gamma} \cdot \gamma^{\alpha + \beta} \) given the condition \( \alpha \gamma = \frac{1}{\beta} \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We start with the condition \( \alpha \gamma = \frac{1}{\beta} \). This can be rearranged to express \( \beta \) in terms of \( \alpha \) and \( \gamma \):
\[ \beta = \frac{1}{\alpha \gamma} \]

2. **Assuming Order of Variables**:
We can assume without loss of generality that \( \alpha \geq \beta \geq \gamma \). This helps us analyze the behavior of the variables under the given constraints.

3. **Analyzing the Variables**:
From the condition \( \alpha \gamma = \frac{1}{\beta} \), we can infer:
- Since \( \beta \) is positive, \( \alpha \) must be greater than or equal to 1 (to keep \( \beta \) positive).
- \( \gamma \) must be less than or equal to 1 (to satisfy the equation).

4. **Rewriting the Expression**:
We rewrite the expression we want to maximize:
\[ E = \alpha^{\beta + \gamma} \cdot \beta^{\beta + \gamma} \cdot \gamma^{\alpha + \beta} \]

5. **Using AM-GM Inequality**:
To find the maximum value, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality. We consider the terms:
\[ \frac{\beta + \gamma + \alpha + \beta + \gamma + \alpha + \beta + \gamma}{3} \geq \sqrt[3]{\alpha^{\beta + \gamma} \cdot \beta^{\beta + \gamma} \cdot \gamma^{\alpha + \beta}} \]
This implies:
\[ \frac{2(\alpha + \beta + \gamma)}{3} \geq \sqrt[3]{E} \]

6. **Finding Maximum Value**:
Given the constraints on \( \alpha \) and \( \gamma \), we can find that the maximum value of \( E \) occurs when \( \alpha = 1 \), \( \beta = 1 \), and \( \gamma = 1 \). Substituting these values:
\[ E = 1^{1 + 1} \cdot 1^{1 + 1} \cdot 1^{1 + 1} = 1 \]

7. **Conclusion**:
Therefore, the maximum value of the expression \( \alpha^{\beta + \gamma} \cdot \beta^{\beta + \gamma} \cdot \gamma^{\alpha + \beta} \) is:
\[ \boxed{1} \]