Chapter 4.1, Problem 55E

For a function f and constant k ≠ 0, we have

$\begin{array}{l}{f}^{\prime }(x)=-k^{2}x{e}^{-0.5x^2/k^2}\\ f^{″}(x)=(x^2-k^2)e^{-0.5x^2/k^2}.\end{array}$

1. (a) What is the only critical point of f?
2. (b) How many inflection points does the graph of f have?
3. (c) Is the critical point of f a local maximum, local minimum, or neither?