5.2 - SECOND DERIVATIVE AND CONCAVITY
I.
The second derivative of
y  f (x) is the derivative of the first derivative and is denoted by
y  f (x) . Similarly, the third derivative is the derivative of the second derivative, denoted by
y  f (x) . The nth derivative of f (x) is denoted f (n) (x) .
Example: Find the second derivative of f ( x) 
9  x2 .
Example: If s(t)  2t 3  7t 2  4t  1 is the position of a moving object at time t, where s(t) is measured
in feet and t is measured in seconds, find:
(a) the velocity at time t.
(b) the acceleration at the times when velocity is zero.
Example: Find
f (6) (x) if f (x)  2x4  3x3  4x2  x  6 .
II.
Partition numbers from the second derivative are possible inflection points (change of concavity):
1.
f 
f (x)
+
=0
or
2.
f (x)
is undefined
f 
 f is concave up
(-)
 f is concave down
THE SECOND DERIVATIVE TEST
COMPUTE
f (c)
f (c) for EACH local extremum c found from f  .
for f
negative
confirms local maximum at c
positive
confirms local minimum at c
zero
test fails and is inconclusive
Examples: