If `phi(x)=1/(1+e^(-x))` `S=phi(5) +phi(4) +phi(3) + ..+phi(-3) + phi (-4) + phi(-5)` then the value of S is

Options

A. 5
B. `11//2`
C. 6
D. `13//2`

Explanation

To solve the problem, we need to evaluate the sum \( S = \phi(5) + \phi(4) + \phi(3) + \phi(2) + \phi(1) + \phi(0) + \phi(-1) + \phi(-2) + \phi(-3) + \phi(-4) + \phi(-5) \) where \( \phi(x) = \frac{1}{1 + e^{-x}} \).

### Step-by-Step Solution:

1. **Understanding the Function**:
We start with the function \( \phi(x) = \frac{1}{1 + e^{-x}} \). We can also find \( \phi(-x) \):
\[ \phi(-x) = \frac{1}{1 + e^{x}} \]

2. **Finding the Relationship**:
We can add \( \phi(x) \) and \( \phi(-x) \):
\[ \phi(x) + \phi(-x) = \frac{1}{1 + e^{-x}} + \frac{1}{1 + e^{x}} \]
To combine these fractions, we find a common denominator:
\[ = \frac{(1 + e^{x}) + (1 + e^{-x})}{(1 + e^{-x})(1 + e^{x})} \]
Simplifying the numerator:
\[ = \frac{2 + e^{x} + e^{-x}}{(1 + e^{-x})(1 + e^{x})} \]
Noticing that \( e^{x} + e^{-x} = 2\cosh(x) \), we can simplify further, but we only need to know that:
\[ \phi(x) + \phi(-x) = 1 \]

3. **Calculating \( S \)**:
Now, we can pair the terms in \( S \):
\[ S = \phi(5) + \phi(-5) + \phi(4) + \phi(-4) + \phi(3) + \phi(-3) + \phi(2) + \phi(-2) + \phi(1) + \phi(-1) + \phi(0) \]
Using the relationship \( \phi(x) + \phi(-x) = 1 \):
- \( \phi(5) + \phi(-5) = 1 \)
- \( \phi(4) + \phi(-4) = 1 \)
- \( \phi(3) + \phi(-3) = 1 \)
- \( \phi(2) + \phi(-2) = 1 \)
- \( \phi(1) + \phi(-1) = 1 \)
- \( \phi(0) = \frac{1}{1 + 1} = \frac{1}{2} \)

Therefore, we have:
\[ S = 5 \cdot 1 + \frac{1}{2} = 5 + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2} \]

4. **Final Result**:
Thus, the value of \( S \) is:
\[ S = \frac{11}{2} \]