3.2 Concavity and Points of Inflection

Review - What the FIRST derivative tells when zero, positive, and negative

New – What the SECOND derivative tells when zero, positive, and negative.

Concavity

Concave up - f ‘ increasing - f ‘’ positive

Concave down - f ‘ decreasing - f ‘’ negative

Concave up - “holds water”

Concave down - “spills water”

Inflection Point A point where the concavity changes. At an inflection point (c, f(c)) either f ‘’(c) = 0 or f ‘’(c) does not exist.

Where is the function concave up and concave down? Find any inflection points and use number line and test points for f ‘’ on the intervals created by the 2^nd order critical numbers.

#6

#10

SUMMARIZE INFORMATION FROM f ‘ and f ‘’

	If equals 0	If positive	If Negative
1^st Derivative f ’
2^nd Derivative f ‘’

Determine where the function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function.

#16

#46 The first derivative f ‘ is given. From that information, find

a) Where the function f is increasing and decreasing

b) Where the function f is concave up and concave down

c) The x coordinates of the relative extrema and inflection points of f

d) Sketch a possible graph for f(x)

Applications

#54 A company estimates that when x thousand dollars are spent on the marketing of a certain product, Q(x) units of the product will be sold, where for 10 < x < 40. Sketch the graph of Q(x). Where does the graph have an inflection point? What is the significance of the marketing expenditure that corresponds to this point?