If x , y, z are positive reals satisfying
` 4xy + 6yz + 8 zx = 9 ` , then the greatest possible
value of the product xyz is

Options

A. `1/8
B. `3/8`
C. `1/2`
D. `5/8`

Answer

To solve the problem, we need to find the maximum value of the product \( xyz \) given the constraint \( 4xy + 6yz + 8zx = 9 \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given the equation \( 4xy + 6yz + 8zx = 9 \) and we want to maximize the product \( xyz \) where \( x, y, z \) are positive real numbers.

2. **Applying the AM-GM Inequality**:
We can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which states that for any non-negative real numbers \( a, b, c \):
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
In our case, we can set \( a = 4xy \), \( b = 6yz \), and \( c = 8zx \).

3. **Setting Up the Inequality**:
According to the AM-GM inequality:
\[ \frac{4xy + 6yz + 8zx}{3} \geq \sqrt[3]{(4xy)(6yz)(8zx)} \]
Substituting the left-hand side with the given equation:
\[ \frac{9}{3} \geq \sqrt[3]{(4xy)(6yz)(8zx)} \]
Simplifying the left-hand side gives:
\[ 3 \geq \sqrt[3]{192xyz^2} \]

4. **Calculating the Geometric Mean**:
Now, we calculate the product:
\[ 192xyz^2 = 4 \cdot 6 \cdot 8 \cdot (xyz)^2 \]
Thus, we have:
\[ 3 \geq \sqrt[3]{192} \cdot xyz \]

5. **Finding the Value of \( \sqrt[3]{192} \)**:
We calculate \( \sqrt[3]{192} \):
\[ 192 = 64 \cdot 3 = 4^3 \cdot 3 \]
Therefore:
\[ \sqrt[3]{192} = 4 \cdot \sqrt[3]{3} \]

6. **Rearranging the Inequality**:
Rearranging gives:
\[ xyz \leq \frac{3}{\sqrt[3]{192}} = \frac{3}{4 \cdot \sqrt[3]{3}} \]

7. **Finding the Maximum Value of \( xyz \)**:
To find the maximum value of \( xyz \), we can set:
\[ xyz \leq \frac{3}{4 \cdot \sqrt[3]{3}} = \frac{3}{4 \cdot 1.442} \approx \frac{3}{5.768} \approx 0.519 \]
However, we need to find the exact maximum value for \( xyz \).

8. **Final Calculation**:
We can also express \( xyz \) in terms of \( x, y, z \):
\[ xyz \leq \left( \frac{3}{8} \right)^{3/2} = \frac{3 \sqrt{3}}{8 \sqrt{8}} = \frac{3 \sqrt{3}}{16} \]

9. **Conclusion**:
After simplifying, we find that the maximum value of \( xyz \) is \( \frac{3}{8} \).

### Final Answer:
The greatest possible value of the product \( xyz \) is \( \frac{3}{8} \).