A function is differentiable for

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

Problem

A function

f

is differentiable for

0\le x\le 6

, and its derivative is defined by

f'(x)=\begin{cases} 4, & 0\le x<2,\\ 0, & 2\le x<3,\\ -2, & 3\le x\le 6. \end{cases}

The value of the function at

x=0

f(0)=5

. (a) Calculate the exact values of

f(2)

f(3)

, and

f(6)

. Show all steps. (b) Determine the absolute maximum value of

f

on the closed interval

[0,6]

. State both the

x

-coordinate where it occurs and the corresponding function value, and justify your answer. (c) Identify all relative extrema of

f

on the open interval

(0,6)

and classify each as a relative maximum or minimum. Provide a justification based on the behavior of

f'