Analyze the polynomial function
g(x)=x5−5x4
with respect to its critical points and local behavior.
(a) Determine g′(x) and write it in completely factored form.
(b) List all critical points of g and state the multiplicity of each corresponding root of g′(x).
(c) Construct a sign chart for g′(x) and identify the intervals on which g is increasing and the intervals on which it is decreasing.
(d) For each critical point, decide whether g has a local maximum, a local minimum, or a point of inflection with a horizontal tangent. Justify your conclusions using the information from parts (b) and (c).