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 theintervals where f increases and the intervals where fdecreases.
d) Find f′′ (x) and simplify it completely
e) Use the second derivative test to classify the CNs as RelativeMaximizers or Relative Minimizers of f (x). If the secondderivative test fails for any CN, you must explicitly show theformal first derivative test for that particular CN.
f) Find all the Hypercritical Numbers (if any). If there are noHCNs, write “none.” Explain.
g) Construct a sign diagram using the HCN and f ′′(x). Formallywrite the intervals where f is convex and the intervals where f isconcave down.
h) Find the abscissas of the inflection points. If there are noinflection points, write “none” and explain why.
i) Find all relative extrema and all the inflection points. Ifthere are none, write “none” and explain why.
j) Using the information in parts a)-i), sketch the graph of thegiven function and show all relevant points explicitly.
DRAW THE CORRECT GRAPH MANNUALLY. IF YOU ARE MISSING ANY PARTS FROMa) TO i), YOUR GRAPH EARNS ZERO CREDIT BECAUSE THE WHOLE POINT ISTO SHOW GRAPHICALLY WHAT YOU DID ALGEBRAICALLY OR USING CALCULUS INPARTS a)-i).