Let $f(x) = xe^{x^{2}}$. Find: (a) the intervals on which $f$ is increasing, (b) the intervals on which $f$ is decreasing, (c) the open intervals on which $f$ is concave up, (d) the open intervals on which $f$ is concave down, and (e) the $x$-coordinates of all inflection points.

(a) $f$ is increasing on the interval(s)
(b) $f$ is decreasing on the interval(s)
(c) $f$ is concave up on the open interval(s)
(d) $f$ is concave down on the open interval(s)
(e) the $x$ coordinate(s) of the points of inflection are

Notes: In the first four boxes, your answer should either be a single interval, such as [0,1), a comma separated list of intervals, such as (-inf, 2), (3,4], or the word "none".

In the last box, your answer should be a comma separated list of $x$ values or the word "none".