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.