The polynomial function is defined by

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

Problem

The polynomial function

f

is defined by

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

. Consider the point

x=2

. (a) Verify that

x=2

is a critical point of

f

. Show all work. (b) Calculate

f''(2)

. Express your answer exactly. (c) Using the value found in part (b), describe the concavity of

f

x=2

and determine whether

f

has a local maximum, a local minimum, or neither at this point. Justify your conclusion.