Consider the function f(x)=2x3−6x2−18x+54\displaystyle {f{{\left({x}\right)}}}={2}{x}^{{3}}-{6}{x}^{{2}}-{18}{x}+{54}.

  1. Find the first derivative. f′(x)=\displaystyle {f}'{\left({x}\right)}=  
  2. List any critical values.
  3. Identify intervals of increase.
  4. Identify intervals of decrease.
  5. Find the second derivaitve. f′′(x)=\displaystyle {f}{''}{\left({x}\right)}=  
  6. Use parts b through e to identify maximums as points.
  7. Use parts b through e to identify minimumns as points.
  8. Use the second derivative to identify intervals where f(x)\displaystyle {f{{\left({x}\right)}}} is concave up.  
  9. Use the second derivative to identify intervals where f(x)\displaystyle {f{{\left({x}\right)}}} is concave down.  
  10. Use the second derivative to find any inflection points.
  11. Upload a sketch of the function based on the information above and the intercepts.

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