Question

As $x$ increased from $-\infty$ to $\infty$ the function $f\left(x\right)=\frac{e^x}{1+e^x}$

• A. monotonically increases
• B. monotonically decreases
• C. increases to a maximum value and then decreases
• D. decreases to a minimum value and then increases

Answer 1

The correct option is A monotonically increases

$f\left(x\right)=\frac{e^x}{1+e^x}$
$f\left(x\right)=\frac{(1+e^x)e^x-e^x\left(e^x\right)}{(1+e^x{)}^2}$
$f^{\prime }\left(x\right)=\frac{e^x}{(1+e^x{)}^2}>0$
$\Rightarrow f\left(x\right)$ is monotonically increasing function.