If x,y,z are positive integers, then the value of the expression `(x+y)(y+z)(z+x)` is

Options

A. `a) =8xyz`
B. b) >8xyz
C. c) `lt8xyz`
D. d) `=4xyz`

Answer

To solve the problem, we need to analyze the expression \((x+y)(y+z)(z+x)\) where \(x\), \(y\), and \(z\) are positive integers. We will use the Arithmetic Mean-Geometric Mean (AM-GM) inequality to derive the required result.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We have the expression \((x+y)(y+z)(z+x)\). Our goal is to find a relationship between this expression and the product \(xyz\).

2. **Applying the AM-GM Inequality**:
The AM-GM inequality states that for any non-negative numbers \(a\) and \(b\):
\[ \frac{a+b}{2} \geq \sqrt{ab} \]
We can apply this to each pair of variables:
- For \(x\) and \(y\):
\[ \frac{x+y}{2} \geq \sqrt{xy} \]
- For \(y\) and \(z\):
\[ \frac{y+z}{2} \geq \sqrt{yz} \]
- For \(z\) and \(x\):
\[ \frac{z+x}{2} \geq \sqrt{zx} \]

3. **Multiplying the Inequalities**:
Now, we multiply these three inequalities together:
\[ \left(\frac{x+y}{2}\right) \left(\frac{y+z}{2}\right) \left(\frac{z+x}{2}\right) \geq \sqrt{xy} \cdot \sqrt{yz} \cdot \sqrt{zx} \]
This simplifies to:
\[ \frac{(x+y)(y+z)(z+x)}{8} \geq \sqrt{(xyz)^2} \]
Which means:
\[ \frac{(x+y)(y+z)(z+x)}{8} \geq xyz \]

4. **Rearranging the Inequality**:
Rearranging the above inequality gives us:
\[ (x+y)(y+z)(z+x) \geq 8xyz \]

5. **Conclusion**:
Since \(x\), \(y\), and \(z\) are positive integers, the expression \((x+y)(y+z)(z+x)\) is always greater than or equal to \(8xyz\). Therefore, the value of the expression \((x+y)(y+z)(z+x)\) is greater than \(8xyz\).

### Final Answer:
The correct option is **(b) greater than \(8xyz\)**.