4-3 The Second Derivative Test
What does the 2nd derivative f  tell us about f ?
Definition: Concavity
Suppose that f is differentiable on an interval I.
f is said to be concave up on I if f  is
increasing on I.
 f is said to be concave down on I if f  is
decreasing on I.

Theorem:
Suppose that
f  exists on an interval I
 If f ( x)  0 on I, then f is concave up on I
 If f ( x)  0 on I, then f is concave down on I
Definition: Inflection Point
Suppose f is continous on an open interval I
The point (c, f (c)) is called an inflection point if f
changes from concave up to concave down (or from
concave down to concave up) at point c
Determine the inflection points and the intervals on
which
4
3
f ( x)  x  4x
1
3
is concave up and concave down.
Second Derivative Test for Local Extrema
Suppose f is continuous on the interval (a, b) and
c(a,b) is a critical number for which f (c)  0 .
 If f (c)  0 , then f (c) is a local maximum.
 If f (c)  0 , then f (c) is a local minimum.
 If f (c)  0 or f (c) is undefined, then we can
conclude nothing.
Determine the local extrema of
4
3
f ( x)  x  4x
1
3
If c is a critical number, but f (c)  0 or undefined,
you can tell nothing about local extrema.
Ex.
f (x)  x4
f (x)  4x3
f (x) 12x2
f ( x)  x4
f ( x) 4x3
f ( x) 12x2
f ( x)  x3
f (x)  3x2
f (x)  6x
Note: all three graphs have
 a critical number at c=0
 f (0)  0
If we look at the graphs:
The graph may have a local minimum, local
maximum or neither
Some Notes on the Second Derivative Test
 The second derivative test is nice to use because, in
most cases, it is easier to use than the first derivative
test
o Find the critical numbers c for which fʹ(x)=0
o Determine the sign of fʹʹ by substituting each c
into fʹʹ
However, there are some restrictions:
 It cannot be used for critical numbers for which fʹ is
undefined
 It cannot be use if fʹ(c)=0 and fʹʹ(c)=0
In either of these cases, we say
“The Second Derivative Test Fails”
 If the 2nd Derivative Test fails, you need to use the 1st
Derivative test to determine whether is a local
extremum or not.