5.3:Higher Order Derivatives, Concavity and the 2nd Derivative Test

Objectives:•To find Higher Order Derivatives•To use the second derivative to test for concavity•To use the 2nd Derivative Test to find relative extrema

If a function’s derivative is f’, the derivative of f’, if it exists, is the second derivative, f’’. You can take 3rd, 4th,5th, etc. derivative

NotationsSecond Derivative:

Third Derivative:

For n> 4, the nth derivative is written f(n)(x)

)(,),('' 2

2

2

xfDdx

ydxf x

3

3

),('''dx

ydxf

1. Find f(4)(x). 10764)( 234 xxxxxf

2. Find f’’(0). xxxxf 23 125)(

Asking to find the 4th derivative of f(x):

24)(

2424)('''

122412)(''

712124)('

)4(

2

23

xf

xxf

xxxf

xxxxf

Asking to find the 2nd derivative and evaluate for x = 0:

24)0(''

2430)(''

2415)(' 2

f

xxf

xxxf

Find f’’(x).

1. 22 7)( xxf 2. x

xxf

1

)(

2812

2848

4)7()2(4)(''

)7(4)2)(7(2)('

:

2

22

2

22

x

xx

xxxxf

xxxxxf

Answer

2

2

1

1

1

)1(1)1()('

x

x

xxxf

For f ‘’(x), easiest to bring up with a negative exponent:

3

3

2

1

2)(''

12)(''

11)('

xxf

xxf

xxf

Find f’’’(x).

2

3)(

x

xxf

44

3

2

22

2

2

36)2(36)('''

)2(12)(''

)2(6)('

2

6

2

363

2

1)3(3)2()('

xxxf

xxf

xxf

xx

xx

x

xxxf

Concavity of a Graph How the curve is turning, shape of the graph

Determined by finding the 2nd derivative

Rate of change of the first derivative

Concave Up: y’ is increasing, graph is “smiling”, cup or bowl Concave Down: y’ is decreasing, graph is “frowning”, arch Inflection point: where a function changes concavity

f’’ = 0 or f’’ does not exist here

Precise Definition of Concave Up and Down

A graph is Concave Up on an interval (a,b) if the graph lies above its tangent line at each point in (a,b)

A graph is Concave Down on an interval (a,b) if graph lies below its tangent line at each point in (a,b)

At inflection points, the graph crosses the tangent line

Test for Concavity

• f’ and f’’ need to exist at all point in an interval (a,b)• Graph is concave up where f’’(x) > 0 for all points in

(a,b)• Graph is concave down where f’’(x) < 0 for all points

in (a,b)

Find inflection points and test on a number line. Pick x-values on either side of inflection points to tell whether f’’ is > 0 or < 0

Find the open intervals where the functions are concave up or concave down. Find any inflection points.

1. 34 4)( xxxf

3

6)(

2 x

xf

3

5

3

8

4)( xxxf

Second Derivative Test for Relative Extrema

Let f’’(x) exist on some open interval containing c, and let f’(c) = 0.

1. If f’’(c) > 0, then f(c) is a relative minimum2. If f’’(c) < 0, then f(c) is a relative maximum3. If f’’(c) = 0 or f’’(c) does not exist, use 1st

derivative test

Find all relative extrema using the

2nd Derivative Test.1. 2. 133)( 23 xxxf 3

5

3

8

)( xxxf

If a function describes the position of an object along a straight line at time t:

s(t) = positions’(t) = v(t) = velocity (can be + or - )s’’(t) = v’(t) = a(t) = acceleration

If v(t) and a(t) are the same sign, object is speeding up

If v(t) and a(t) are opposite signs, object is slowing down

Suppose a car is moving in a straight line, with its position from a starting point (in ft) at time t (in sec) is given by s(t)=t3-2t2-7t+9

a.) Find where the car is moving forwards and backwards.

b.) When is the car speeding up and slowing down?