The logistic function

is used to study the spread of infectious diseases such as viruses.

1. Show that the curve is increasing for all values of $x.$

2. Find the range of values of $x$ for which the curve is convex, and give the coordinates of the point of inflection of the curve.

3. Show that

   $$f^{\prime }(x)=f(x)f(-x)$$

   and hence express $f^{″}(x)$ in terms of $f(x)$ and $f(-x).$