Let x, y be positive real numbers and m, n be positive integers, The maximum value of the expression
`(x^(m)y^(n))/((1+x^(2m))(1+y^(2n)))` is

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(m+n)/(6mn)`
D. 1

Answer

To find the maximum value of the expression

\[ E = \frac{x^m y^n}{(1+x^{2m})(1+y^{2n})} \]

where \(x, y\) are positive real numbers and \(m, n\) are positive integers, we can utilize the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Apply AM-GM Inequality**:
We start by applying the AM-GM inequality to the terms \(1\) and \(x^{2m}\):
\[ \frac{1 + x^{2m}}{2} \geq \sqrt{1 \cdot x^{2m}} = x^m \]
This implies:
\[ 1 + x^{2m} \geq 2x^m \]

2. **Apply AM-GM Inequality Again**:
Similarly, apply the AM-GM inequality to the terms \(1\) and \(y^{2n}\):
\[ \frac{1 + y^{2n}}{2} \geq \sqrt{1 \cdot y^{2n}} = y^n \]
This implies:
\[ 1 + y^{2n} \geq 2y^n \]

3. **Multiply the Results**:
Now, we multiply the two inequalities obtained:
\[ (1 + x^{2m})(1 + y^{2n}) \geq (2x^m)(2y^n) = 4x^m y^n \]

4. **Rearranging the Expression**:
From the above inequality, we can rearrange it to find an upper bound for our expression \(E\):
\[ E = \frac{x^m y^n}{(1+x^{2m})(1+y^{2n})} \leq \frac{x^m y^n}{4x^m y^n} = \frac{1}{4} \]

5. **Conclusion**:
Thus, the maximum value of the expression \(E\) is:
\[ \boxed{\frac{1}{4}} \]