f(x)=(1)/(x^(2)+1)

Question

1. determine the intervals on which f is concave up and down
2. determine (if any) all inflection points of f.

Answer 1

f(x)=f(-x) so f(x) is an even function.

The maximum value of f(x) is f(0)=1, the point (0,1).

(1) function f is concave down over the whole domain (-∞,∞) with a maximum at (0,1). But for x in (-∞,0) the gradient is increasing and for x in (0,∞) the gradient is decreasing.

(2) there is no point of inflexion.