Let `alpha, beta, gamma` and `delta` be four positive real numbers such that their product is unity, then the least value of `(1 + alpha)(1+beta)(1+gamma)(1+delta)` is

Options

A. 6
B. 16
C. 0
D. 32

Answer

To find the least value of \( (1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta) \) given that \( \alpha, \beta, \gamma, \delta \) are positive real numbers such that their product is unity (\( \alpha \beta \gamma \delta = 1 \)), we can use the AM-GM inequality.

### Step-by-Step Solution:

1. **Apply the AM-GM Inequality**:
The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any non-negative real numbers \( x_1, x_2, \ldots, x_n \):
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
with equality when \( x_1 = x_2 = \ldots = x_n \).

2. **Set Up the Terms**:
We can rewrite \( (1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta) \) as:
\[ 1 + \alpha + \beta + \gamma + \delta + (\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta) + (\alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta) + \alpha\beta\gamma\delta \]

3. **Use the Condition \( \alpha \beta \gamma \delta = 1 \)**:
Since \( \alpha \beta \gamma \delta = 1 \), we can express the product \( \alpha\beta\gamma\delta \) as \( 1 \).

4. **Apply AM-GM to Each Term**:
- For \( \alpha, \beta, \gamma, \delta \):
\[ \frac{\alpha + \beta + \gamma + \delta}{4} \geq \sqrt[4]{\alpha \beta \gamma \delta} = \sqrt[4]{1} = 1 \]
Hence, \( \alpha + \beta + \gamma + \delta \geq 4 \).

- For \( 1 + \alpha, 1 + \beta, 1 + \gamma, 1 + \delta \):
\[ \frac{(1 + \alpha) + (1 + \beta) + (1 + \gamma) + (1 + \delta)}{4} \geq \sqrt[4]{(1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta)} \]
This leads to:
\[ \frac{4 + \alpha + \beta + \gamma + \delta}{4} \geq \sqrt[4]{(1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta)} \]

5. **Combine Results**:
From the previous steps, we have:
\[ \frac{4 + \alpha + \beta + \gamma + \delta}{4} \geq \sqrt[4]{(1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta)} \]
Since \( \alpha + \beta + \gamma + \delta \geq 4 \), we can substitute:
\[ \frac{4 + 4}{4} = 2 \geq \sqrt[4]{(1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta)} \]

6. **Raise Both Sides to the Fourth Power**:
\[ 16 \geq (1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta) \]

7. **Conclusion**:
The least value of \( (1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta) \) is \( 16 \), which occurs when \( \alpha = \beta = \gamma = \delta = 1 \).

### Final Answer:
The least value of \( (1 + \alpha)(1 + \beta)(1 + \gamma)(1 + \delta) \) is \( \boxed{16} \).