Consider the function defined by .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider the function

f

defined by

f(x)=x^{3}-6x^{2}+9x+2

. (1) Part 1: (a) Calculate

f'(x)

and determine all critical numbers of

f

. (2) 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. (3) Part 3: (c) Verify the classifications in part (b) by evaluating

f''(x)

at each critical number and interpreting the result.