For the function `f(x)=(e^x)/(1+e^x),` which of the following hold good? `f` is monotonic in its entire domain. Maximum of `f` is not attained even though `f` is bounded `f` has a point of inflection. `f` has one asymptote.

Options

A. f is monotonic in its entire domain
B. maximum of f is not attained even thought
C. f is bounded
D. f ahs a point of inflection

Answer

1,2,3

Explanation

`f(x)=(e^(x))/(1+e^(x))`
`therefore f(X)=e^(x)(1+e^(x)-e^(x)e^(x))/(1+e^(x)) =(e^(x))/(1+xe^(x))^(2)gt0forallx in R`
Thus f(x) is an increasing funciton
Also `underset(xrarr-oo)lim(e^(x))/(1+e^(x))=0` and `underset(xrarroo)lix(e^(x))/(1+e^(x))=underset(xrarroo)lim(1)/(1+(1)/(e^(x))=1`
Hene the graph of `f(X)=(e^(x))/(1+e^(x))` is as shown

Also let `f(X)=e^(x)1+e^(x)^(2)-2(1+e^(x)e^(x)e^(x))/(1+e^(x))a^(4)=0`
or `e^(x) =1`
or x=0 which is point of inflection
Thus x =0 is the inflection point and f is bounded in (0,1)
There is no maxima and f has two asymptotes