Section 5.2 – Applications of the Second Derivative

THE SECOND DERIVATIVE TEST FOR MAX/MIN

If f ' a 0 and , then f a is a .

I

f " a 0 minimum

ff f ' a 0 and , then f a is a" a 0 maximum.

Theory First….POINT OF INFLECTION

If f" a 0 and f " a changes from pos to neg or neg to pos,

a point of inflection exists at x = a

TEST FOR CONCAVITY

If f" a f is concave at x = a

If f" a f is con

0 up

0 docave at xwn = a

3 2If f x x 5x 3x, then the graph of f is decreasing

and concave down on the interval

1 1 2 1 5 5A. 0, B. , C. , D. ,3 E. 3,

3 3 3 3 3 3

2f ' x 3x 10x 3

0 3x 1 x 3

1x , 3

3

1

33

+ _ +

X X

f " x 6x 10

0 6x 10

5x

3

5

3

_ +

2

Suppose a function f is defined so that it has derivatives

f ' x x 1 x and f " x x 2 3x . Over which interval

is the graph of f both increasing and concave up?

2 2A. x 0 B. 0 x C. x 1 D. x 1 E. none of thes

3 3

e

0 1

+ + _

X

0 2

3

_ + _

3 2Find the maximum value of f x 2x 3x 12x 4 on

the closed interval [0, 2]. Justify your answer.

2f ' x 6x 6x 12

0 6 x 2 x 1

x 2,1

f " x 12x 6

f " 2 0

f " 1 0

Since f ‘ (-2) = 0 and f “ (-2) < 0 there is a relative max at x = -2,but x = -2 is not on the interval [0, 2]

f 0 4 f 2 8 Since f(2) > f(0), the abs max occurs at x = 2

The maximum value is 8

2 3

Find the x-coordinates of the point(s) of inflection of the

function f if f ' x x 2 x 7 . Justify your answer.

3 2 2f " x 2 x 2 x 7 3 x 7 x 2

2f " x x 2 x 7 2 x 7 3 x 2

20 x 2 x 7 5x 20

2 4 7

+ _ + +

x = 2 since f “ (x) changes from pos to neg.

x = 4 since f “ (x) changes from neg to pos.

2

Which of the following are true about the function f if its

derivative is defined by f ' x x 1 4 x ?

I. f is decreasing for all x < 4.

II. f has a local maximum at x = 1.

III. f is concave up for all 1

< x < 3.

A. I only B. II only C. III only D. II and III only E. I, II, III

1 4

+ + _

XX

2f " x 2 x 1 4 x 1 x 1

f " x x 1 2 4 x 1 x 1

0 x 1 9 3x x 1, 3

f " 2 0

2

The number of inflection points for the graph of

y 2x cos x in the interval [0, 5] is:

A. 6 B. 7 C. 8 D. 9 E. 10

X

CALCULATOR REQUIRED

CALCULATOR REQUIRED

x 2

Suppose the continuous function f is defined on the closed

interval [0, 3] such that its derivative f ' is defined by

f ' x e sin x 1. Which of the following are true about

the graph of f?

I. f has exa

ctly one relative maximum point.

II. f has two relative minimum points.

III. f has two inflection points.

A. I only B. II only C. III only D. I and II only E. I, II, III

dy

dx

2

2

d y

dx

2

The position at time t > 0 of a particle moving on the x-axis

is given by x t sin tcos t. At the first instant when the

acceleration is 1 unit per sec , the particle has velocity

A. -1 units per sec

B.

-0.866 units per sec

C. 0 units per sec

D. 0.866 units per sec

E. 1 unit per sec

x t sin tcos t

x t sin t1

t22

cos

1x t sin2t

2

v t cos2t a t 2sin2t

1 2sin2t1

sin2t2

c3

2os2t

CALCULATOR REQUIRED

4 5x x

Consider the function f x . Find the value of x for2 10

which the derivative of f attains its maximum. Justify your answer.

4

3 xf ' x 2x

2

2 3f " x 6x 2x

20 2x 3 x x 0, 3

2f "' x 12x 6x

f "' x 6x 2 x

0 0f "' 6 2 0 0

3 3f "' 6 2 3 0

Since f " 3 0 and f "' 3 0, the derivative of f achieves

a maximum at x = 3

NO CALCULATOR

2Which of the following is true about the graph of f x ln x 4

in the interval -2, 2 ?

A. f is increasing

B. f attains a relative minimum at (0, 0)

C. f has a range of all real numbers

D. f is concave down

E. f has an asymptote at x = 0

2 2On 2, 2 , f x ln x 4 ln 4 x

2

2xf ' x

4 x

X

2

22

2 4 x 2x 2xf " x

4 x

22

2 22 2

2 x 42x 8

4 x 4 x

X

X

3 2Let f x x px qx.

a Find the values of p and q so that f 1 8 and f ' 1 12

b Find the values of p so that the graph of f changes concavity

at x = 2

(c) Under what conditions of p and q will the graph of

f be

increasing everywhere? (BC only)

3 2Let f x x px qx.

a Find the values of p and q so that f 1 8 and f ' 1 12

p 81 qf 1 2f ' x 3x 2px q

3 2p q 2' 1f 1

p q 7

2p q 9p 2, q 5

3 2Let f x x px qx.

b Find the values of p so that the graph of f changes concavity

at x = 2

2f ' x 3x 2px q

f " x 6x 2p

f " 2 6 2 2p

0 12 2p

p 6

3 2Let f x x px qx.

(c) Under what conditions of p and q will the graph of f be

increasing everywhere?

2f ' x 3x 2px q

22p 4 3 q 0

24p 12q 0 24p 12q2p 3q

4 2Let f x x ax b. The graph of f has a relative maximum

at (0, 1) and an inflection point when x = 1. Find the values of

a and b.

3f ' x 4x 2ax

2f " x 12x 2a

20 12 1 2a a 6

4 2f x x 6x b

4 21 0 6 0 b

b 1

CALCULATOR REQUIRED

2

dy cos xIf f(x) is defined on - x and , which of the

dx x 1following statements about the graph of y f x is true?

A. The graph has no relative extremum.

B. The graph has one point of inflection and two

relative extrema.

C. The graph has two points of inflection and one relative extremum.

D. The graph has two points of inflection and two relative extrema.

E. The graph has three points of inflection and two relative extrema.

X

X

dy

dx

2

2

d y

dx