Consider the polynomial function

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

Problem

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

. Answer each of the following questions completely, showing all analytical work. (1) Calculate

f'(x)

and

f''(x)

. (2) Solve

f''(x)=0

to obtain all critical numbers associated with possible changes in concavity. (3) Determine the sign of

f''(x)

on each interval determined in part (b), and classify the concavity of

f

on those intervals. (4) Verify each inflection point of

f

by confirming that the concavity changes. State the

x

–coordinate(s) and the corresponding point(s)

(x,f(x))