Thoroughly investigate the graph of $y = xe^{x}$. Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify any point(s) of inflection and intercepts and provide graphs of $y = f ′\left(x\right)$ and $y = f ″\left(x\right)$. Be sure to justify all statements both graphically and analytically.