Mathematics

Problem

Let

g(x)=\dfrac{e^{x}}{1+e^{x}}

x\in\mathbb R

. Answer the following: a. (i) Find

g'(x)

. b. (ii) Show that

g'(x)>0

for all

x\in\mathbb R

. c. (iii) What conclusion can you draw about the existence of local extrema of

g

\mathbb R

? Give reasons. d. (iv) Evaluate

\displaystyle \lim_{x\to -\infty}g(x)

and

\displaystyle \lim_{x\to \infty}g(x)

. Using these limits together with your result from part (ii), comment on the range of

g