Consider the function defined for all real numbers by .

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

Problem

Consider the function

f

defined for all real numbers by

f(x)=|x^{2}-4|

. (a) Rewrite

f

as an explicit piecewise function that does not involve the absolute value symbol. (b) Determine a complete piecewise expression for the derivative

f'(x)

, including any values of

x

where the derivative does not exist. Show all supporting limit calculations. (c) On the axes provided in your answer booklet, sketch the graph of

y=f'(x)

for

-5\le x\le5

. Clearly label the

x

-intercepts and indicate any points where

f'

is not defined. (d) Using your work above, describe the intervals on which

f

is concave up and concave down. Explain how the shape of your sketch in part (c) leads to your conclusion.