Consider the function f defined by
f(x)=x3−6x2+9x+2.
a. Part 1: (a) Calculate f′(x) and determine all critical numbers of f.
b. Part 2: (b) For each critical number found in part (a), determine whether f has a local maximum, a local minimum, or neither at that x–value. Clearly state both the x–coordinate and the corresponding function value of each extremum.
c. Part 3: (c) Verify the classifications in part (b) by evaluating f′′(x) at each critical number and interpreting the result.