The derivative of a continuous function is piecewise linear on with these properties

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

Problem

The derivative

h'

of a continuous function

h

is piecewise linear on

0\le x\le7

with these properties. • On

0\le x\le2

h'(x)

decreases linearly from

3

x=0

-1

x=2

. • On

2\le x\le5

h'(x)

increases linearly from

-1

x=2

4

x=5

. • On

5\le x\le7

h'(x)

decreases linearly from

4

x=5

0

x=7

. The value of the original function is

h(0)=2

. (a) Determine the intervals on which

h

is concave up and the intervals on which

h

is concave down for

0<x<7

. Provide clear justification based on the behavior of

h'

. (b) Identify all points of inflection of

h

in the interval

0<x<7

, giving both the

x

–coordinate and a brief justification for each point selected. (c) Using the information from parts (a) and (b) together with the sign of

h'

, create a carefully labeled sketch or detailed verbal description of one possible graph of

h

0\le x\le7

that satisfies all given conditions, including the initial value

h(0)=2

. Indicate all critical points and inflection points on your representation.