Mathematics

Problem

Let

h

be a twice-differentiable function whose first derivative

h'

is defined for

-8\le x\le 9

. A detailed verbal description of the graph of

h'

is given below. Unless stated otherwise,

h'

is continuous and differentiable everywhere on

(-8,9)

. •

h'(x)=0

x=-6,-2,4,8

and only at these

x

-values. • On

(-8,-6)

h'(x)>0

and

h'

is increasing. • On

(-6,-2)

h'(x)<0

and

h'

is decreasing. • On

(-2,1)

h'(x)>0

and

h'

is decreasing. • On

(1,4)

h'(x)>0

and

h'

is increasing. • On

(4,8)

h'(x)<0

and

h'

is increasing. • On

(8,9)

h'(x)>0

and

h'

is decreasing. In addition,

h(0)=3

. a. Create a complete sign chart for

h'

on the interval

[-8,9]

. Clearly indicate the sign of

h'

on each subinterval determined by the critical points listed above. b. Based on your sign chart, determine all intervals where

h

is increasing and all intervals where

h

is decreasing. Provide interval notation and justify each interval by referencing your sign chart. c. Identify the

x

–coordinates of all relative extrema of

h

and classify each as a relative maximum or a relative minimum. Justify your classifications using the First Derivative Test. d. Sketch a possible graph of

h

[-8,9]

that is consistent with all given information, including the condition

h(0)=3

. Label the relative extrema on your sketch. (A precise scale is not required, but qualitative features must be accurate.)

Intuitive explanation

The key is to distinguish between where $h$ itself is increasing or decreasing (determined by the sign of $h'$ ) and where $h$ is concave up or down (determined by the sign of $h''$ , which tells you whether $h'$ is increasing or decreasing). Use the given information about $h'$ 's zeros and monotonicity to identify local extrema of $h$ , inflection points, and the overall shape of $h$ . Then integrate the behavior piece by piece, anchoring your sketch with the known point $h(0)=3$ .

Intuitive explanation

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