If `a,bgt0` then the maximum value of `(a^(3)b)/((a+b)^(4)),` is

Options

A. `(81)/(512)`
B. `(27)/(256)`
C. `(27)/(512)`
D. `(81)/(256)`

Explanation

To find the maximum value of the expression \(\frac{a^3 b}{(a+b)^4}\) for \(a, b > 0\), we can use the AM-GM inequality.

### Step-by-Step Solution:

1. **Apply AM-GM Inequality**:
We want to apply the AM-GM inequality to the terms \(a, a, a, b\). According to the AM-GM inequality:
\[ \frac{a + a + a + b}{4} \geq \sqrt[4]{a^3 b} \]
This simplifies to:
\[ \frac{3a + b}{4} \geq \sqrt[4]{a^3 b} \]

**Hint**: Remember that AM-GM states that the arithmetic mean is greater than or equal to the geometric mean.

2. **Raise Both Sides to the Power of 4**:
To eliminate the fourth root, raise both sides to the power of 4:
\[ \left(\frac{3a + b}{4}\right)^4 \geq a^3 b \]

**Hint**: Raising both sides of an inequality to an even power preserves the inequality.

3. **Rearranging the Inequality**:
Now, we can express \(a^3 b\) in terms of \(3a + b\):
\[ a^3 b \leq \left(\frac{3a + b}{4}\right)^4 \]

4. **Substituting Back into the Original Expression**:
We want to find the maximum of:
\[ \frac{a^3 b}{(a+b)^4} \]
From the previous step, we know:
\[ a^3 b \leq \left(\frac{3a + b}{4}\right)^4 \]
Therefore,
\[ \frac{a^3 b}{(a+b)^4} \leq \frac{\left(\frac{3a + b}{4}\right)^4}{(a+b)^4} \]

5. **Finding the Maximum Value**:
To find the maximum value, we need to consider the case when equality holds in the AM-GM inequality, which occurs when all terms are equal:
\[ a = a = a = b \implies 3a = b \implies b = 3a \]

Substituting \(b = 3a\) into the expression:
\[ a + b = a + 3a = 4a \]
Thus,
\[ \frac{a^3 (3a)}{(4a)^4} = \frac{3a^4}{256a^4} = \frac{3}{256} \]

6. **Final Calculation**:
The maximum value of \(\frac{a^3 b}{(a+b)^4}\) occurs at:
\[ \frac{27}{256} \]

Therefore, the maximum value of \(\frac{a^3 b}{(a+b)^4}\) is \(\frac{27}{256}\).

### Final Answer:
The maximum value of \(\frac{a^3 b}{(a+b)^4}\) is \(\frac{27}{256}\).