If a + b + c + d = 1 , then the maximum value of
`(1 + a) (1 + b)(1 + c) (1 + d)` is

Options

A. 1
B. `((1)/(2))^(3)`
C. `((3)/(4))^(3)`
D. `((5)/(4))^(4)`

Answer

To find the maximum value of the expression \( (1 + a)(1 + b)(1 + c)(1 + d) \) given that \( a + b + c + d = 1 \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ (1 + a)(1 + b)(1 + c)(1 + d) \]

2. **Apply AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative numbers \( x_1, x_2, x_3, x_4 \):
\[ \frac{x_1 + x_2 + x_3 + x_4}{4} \geq \sqrt[4]{x_1 x_2 x_3 x_4} \]
We can apply this to \( 1 + a, 1 + b, 1 + c, 1 + d \).

3. **Calculate the Arithmetic Mean**:
The arithmetic mean of \( 1 + a, 1 + b, 1 + c, 1 + d \) is:
\[ \frac{(1 + a) + (1 + b) + (1 + c) + (1 + d)}{4} = \frac{4 + (a + b + c + d)}{4} = \frac{4 + 1}{4} = \frac{5}{4} \]

4. **Apply AM-GM**:
By AM-GM, we have:
\[ \frac{5}{4} \geq \sqrt[4]{(1 + a)(1 + b)(1 + c)(1 + d)} \]

5. **Raise Both Sides to the Fourth Power**:
Raising both sides to the fourth power gives:
\[ \left(\frac{5}{4}\right)^4 \geq (1 + a)(1 + b)(1 + c)(1 + d) \]

6. **Calculate \( \left(\frac{5}{4}\right)^4 \)**:
Now we compute \( \left(\frac{5}{4}\right)^4 \):
\[ \left(\frac{5}{4}\right)^4 = \frac{625}{256} \]

7. **Conclusion**:
Thus, the maximum value of \( (1 + a)(1 + b)(1 + c)(1 + d) \) is:
\[ \frac{625}{256} \]

### Final Answer:
The maximum value of \( (1 + a)(1 + b)(1 + c)(1 + d) \) is \( \left(\frac{5}{4}\right)^4 = \frac{625}{256} \).