Let for all real .

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

Problem

Let

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

for all real

x

. (1) Calculate

f'(x)

. (2) Determine every critical point of

f

(-\infty,\infty)

. (3) For each critical point, classify it as a relative minimum or relative maximum. Provide clear reasoning based on the change in sign of

f'(x)

. (4) Arrange the relative extrema from lowest to highest

f

-value.