Question

If $\varphi \left(x\right)=\frac{1}{1+e^{-x}}$ then the value of $\varphi \left(5\right)+\varphi \left(4\right)+...+\varphi (-3)+\varphi (-4)+\varphi (-5)$

• A. $5$
• B. $\frac{9}{2}$
• C. $\frac{11}{2}$
• D. none of these

Answer 1

The correct option is C $\frac{11}{2}$

Given $\varphi \left(x\right)=\frac{1}{1+e^{-x}}$

$\Rightarrow \varphi \left(x\right)=\frac{e^x}{e^x+1}$ .....(1)

Also, $\varphi (-x)=\frac{1}{1+e^x}$ .....(2)

Adding (1) and (2),

$\varphi \left(x\right)+\varphi (-x)=1$
Now, $\varphi \left(5\right)+\varphi \left(4\right)+...+\varphi (-3)+\varphi (-4)+\varphi (-5)$

Pairing the terms, we get

$\varphi \left(5\right)+\varphi (-5)+\varphi \left(4\right)+\varphi (-4)...+\varphi \left(1\right)+\varphi (-1)+\varphi \left(0\right)$

$=5\left(1\right)+\frac{1}{2}$

$=\frac{11}{2}$