Question

Consider the function: f(x) = (x-1)⁴(x+1)

a) Differentiate and simplify completely
b) Find all the Critical Numbers (if any). If there are no CNs, write “none.” Explain why.
c) Construct the correct sign diagram and formally write the intervals where f increases and the intervals where f decreases.
d) Find f′′ (x) and simplify it completely
e) Use the second derivative test to classify the CNs as Relative Maximizers or Relative Minimizers of f (x). If the second derivative test fails for any CN, you must explicitly show the formal first derivative test for that particular CN.
f) Find all the Hypercritical Numbers (if any). If there are no HCNs, write “none.” Explain.
g) Construct a sign diagram using the HCN and f ′′(x). Formally write the intervals where f is convex and the intervals where f is concave down.
h) Find the abscissas of the inflection points. If there are no inflection points, write “none” and explain why.
i) Find all relative extrema and all the inflection points. If there are none, write “none” and explain why.
j) Using the information in parts a)-i), sketch the graph of the given function and show all relevant points explicitly.
DRAW THE CORRECT GRAPH MANNUALLY. IF YOU ARE MISSING ANY PARTS FROM a) TO i), YOUR GRAPH EARNS ZERO CREDIT BECAUSE THE WHOLE POINT IS TO SHOW GRAPHICALLY WHAT YOU DID ALGEBRAICALLY OR USING CALCULUS IN PARTS a)-i).

Answer 1