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
Notations
2
d y
2
Second Derivative: f ' ' ( x ), 2 , D x  f ( x ) 
dx
3
Third Derivative:
f ' ' ' ( x ),
d y
dx
3
For n> 4, the nth derivative is written f(n)(x)
1. Find f(4)(x).
2. Find f’’(0).
f ( x )  x  4 x  6 x  7 x  10
4
3
2
f ( x )  5 x  12 x  x
3
2
Find f’’(x).
1. f ( x )   x  7 
2
2
2. f ( x ) 
x
1 x
Find f’’’(x).
f ( x) 
3x
x2
If a function describes the position of an object
along a straight line at time t:
s(t) = position
s’(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?
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 xvalues 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. f ( x )  x  4 x
4
3
f ( x) 
6
x 3
2
8
5
f ( x)  x 3  4 x 3
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 minimum
2. If f’’(c) < 0, then f(c) is a relative maximum
3. If f’’(c) = 0 or f’’(c) does not exist, use 1st
derivative test
Find all relative extrema using the
2nd Derivative Test.
8
1.
f ( x)  3x  3x  1
3
2
2.
5
f (x)  x 3  x 3