(Long Answer – Conceptual) A function is defined on the real line.

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

Problem

(Long Answer – Conceptual) A function

f

is defined on the real line. (i) State the meaning of a critical point of

f

. (ii) Let

g(x)=|x|

. Verify that

x=0

is a critical point of

g

even though

g'(0)

does not exist. (iii) By examining the values of

g(x)

for

x

just to the left and just to the right of

0

, decide whether

x=0

gives a local maximum, a local minimum, or neither. Give a brief reason for your answer.