If p,q,r be three distinct real numbers, then the least possible value of `(p+q)(q+r)(r+p),` is

Options

A. `gt8pqr`
B. `lt8pqr`
C. `8pqr`
D. none of these

Explanation

To find the least possible value of \((p+q)(q+r)(r+p)\) for three distinct real numbers \(p\), \(q\), and \(r\), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-step Solution:

1. **Apply AM-GM Inequality**:
We will apply the AM-GM inequality to pairs of the sums \(p+q\), \(q+r\), and \(r+p\).

- For \(p\) and \(q\):
\[ p + q \geq 2\sqrt{pq} \]

- For \(q\) and \(r\):
\[ q + r \geq 2\sqrt{qr} \]

- For \(r\) and \(p\):
\[ r + p \geq 2\sqrt{rp} \]

2. **Multiply the Inequalities**:
Now, we multiply the three inequalities together:
\[ (p+q)(q+r)(r+p) \geq (2\sqrt{pq})(2\sqrt{qr})(2\sqrt{rp}) \]

This simplifies to:
\[ (p+q)(q+r)(r+p) \geq 8\sqrt{(pqr)^2} = 8pqr \]

3. **Conclusion**:
Since \(p\), \(q\), and \(r\) are distinct real numbers, the minimum value of \((p+q)(q+r)(r+p)\) occurs when \(pqr\) is minimized. The least possible value of \((p+q)(q+r)(r+p)\) is therefore:
\[ \text{Least possible value} = 8pqr \]

Given that \(p\), \(q\), and \(r\) can take any distinct real values, the minimum value of \(pqr\) can be negative, leading to the conclusion that the least possible value of \((p+q)(q+r)(r+p)\) is \(8\) when \(p\), \(q\), and \(r\) are chosen appropriately.

### Final Answer:
The least possible value of \((p+q)(q+r)(r+p)\) is \(8\).

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