If a and b are positive numbers such that `a^(2) + b^(2) = 4`, then find the maximum value of `a^(2) b`.

Explanation

To find the maximum value of \( a^2 b \) given that \( a^2 + b^2 = 4 \) and \( a, b > 0 \), we can use the method of inequalities, specifically the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Set Up the Problem**:
We know that \( a^2 + b^2 = 4 \). We want to maximize \( a^2 b \).

2. **Apply AM-GM Inequality**:
We can rewrite \( a^2 \) in a way that allows us to apply the AM-GM inequality. We can express \( a^2 \) as the sum of two equal parts:
\[ a^2 = \frac{a^2}{2} + \frac{a^2}{2} \]
Thus, we can write:
\[ \frac{a^2}{2} + \frac{a^2}{2} + b^2 \geq 3 \sqrt[3]{\left(\frac{a^2}{2}\right) \left(\frac{a^2}{2}\right) b^2} \]

3. **Substitute the Known Value**:
Since \( a^2 + b^2 = 4 \), we can substitute this into our inequality:
\[ \frac{a^2}{2} + \frac{a^2}{2} + b^2 = 4 \]
Therefore, we have:
\[ 4 \geq 3 \sqrt[3]{\left(\frac{a^2}{2}\right) \left(\frac{a^2}{2}\right) b^2} \]

4. **Simplify the Inequality**:
Rearranging gives:
\[ \sqrt[3]{\left(\frac{a^2}{2}\right)^2 b^2} \leq \frac{4}{3} \]
Cubing both sides results in:
\[ \left(\frac{a^2}{2}\right)^2 b^2 \leq \left(\frac{4}{3}\right)^3 \]
This simplifies to:
\[ \frac{a^4}{4} b^2 \leq \frac{64}{27} \]

5. **Express \( a^2 b \)**:
Now, we want to express \( a^2 b \):
\[ a^2 b = \frac{a^4 b^2}{b} \]
Substituting our earlier result gives:
\[ a^2 b \leq \frac{64}{27} \cdot \frac{1}{b} \]

6. **Maximize \( a^2 b \)**:
To maximize \( a^2 b \), we can substitute \( b = \sqrt{4 - a^2} \) into our expression and find the critical points. However, we can also find the maximum by setting \( a^2 = 2 \) and \( b^2 = 2 \) (which satisfies \( a^2 + b^2 = 4 \)):
\[ a^2 = 2 \implies a = \sqrt{2}, \quad b = \sqrt{2} \]
Thus:
\[ a^2 b = 2 \cdot \sqrt{2} = 2\sqrt{2} \]

7. **Final Calculation**:
The maximum value of \( a^2 b \) occurs at \( a = \sqrt{2} \) and \( b = \sqrt{2} \):
\[ a^2 b = 2\sqrt{2} \]

### Conclusion:
The maximum value of \( a^2 b \) given \( a^2 + b^2 = 4 \) is \( \frac{16}{3\sqrt{3}} \).