A function is defined and differentiable on the closed interval

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

Problem

A function

g

is defined and differentiable on the closed interval

0\le x\le 9

. The graph of

g'

consists of three straight-line segments that connect the points

(0,-2)

(3,4)

(6,-1)

, and

(9,0)

in the

xy

-plane. (a) Identify all values of

x

[0,9]

at which

g

has a critical point. (b) For each sub-interval determined by the

x

-coordinates from part (a) and the endpoints, determine the sign of

g'(x)

and indicate where

g

is increasing or decreasing. (c) Without calculating any

g

-values, determine and justify the

x

-coordinate(s) of the absolute maximum and the absolute minimum of

g

[0,9]

. (d) Given that

g(0)=5

, compute the absolute maximum value and the absolute minimum value of

g

[0,9]

. Provide supporting work.