Thoroughly investigate the graph of $f(x) = e^{−x^2}$ . Identify all important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection, extrema, intercepts, and provide graphs of $f^\prime(x)$ and $f^{\prime\prime}(x)$. Be sure to justify all statements both graphically and analytically.