Consider the function g(x)=xe−x2 defined for all real numbers.
a. Compute g′(x) and find all values of x at which g′(x)=0. (Show algebraic work.)
b. Create a sign chart for g′(x) on the real number line and determine the intervals on which g is increasing and decreasing.
c. Classify each critical point found in part (a) as a relative minimum or relative maximum and justify your conclusion.
d. Rank the relative extrema from smallest to largest g-value. (Exact values or simplified expressions are acceptable.)