5.2: The Second Derivative
Definition: Given a function y = f (x). We denote by f 00 (x) the second derivative of f (x), which is the
derivative of f 0 (x):
d 0
(f (x)),
f 00 (x) =
dx
and is a function.
Other notations:
d2
d2 y
f
(x)
or
.
dx2
dx2
7
Example 1. Given f (x) = (1 − 3x) 4 .
a) Find f 0 (x).
b) Find f 00 (x).
Given s(t) as a position function of an object moving on a straight line. Then the velocity v(t) is s0 (t),
a function of time t.
The acceleration a(t) is a(t) = v 0 (t) = s00 (t) as the instantaneous rate of change of velocity with
respect to time t.
Example 2. A particle moves along a straight line with equation of motion
s(t) =
t3 3t2
−
− 2t + 1
4
2
where s is measured in meters and t in seconds.
a) Find the acceleration as a function of time.
b) Determine the acceleration when t = 3?
1
Similar question to the previous section, what does f 00 say about f ?
Since f 00 is the derivative of f 0 , we know that if f 00 (x) is positive, then f 0 is an increasing function.
I.e. the slopes of the tangent lines of the graph of y = f (x) increase from left to right. The slope of
this curve becomes progressively larger as x increases and we observe that, as a consequence, the curve
bends upward, and f is said to be concave upward. Notice that in this case the curve always lies above
its tangent.
If f 00 (x) is negative, f 0 is decreasing. Thus, the slopes of the tangent lines of the graph of f decrease
from left to right and the curve bends downward. Thus, this curve is concave downward. Notice that
in this case the curve always lies below its tangent.
Test for Concavity:
• If f 00 (x) > 0 on an interval, then f is concave upward on that interval.
• If f 00 (x) < 0 on an interval, then f is concave downward on that interval.
Example 3. Given the graph of a function
Determine the intervals where the function is
a) concave up.
b) concave down.
2
Example 4. Given the graph of the derivative f 0 .
Determine the intervals where the function is
a) increasing
b) decreasing
c) concave up
d) concave down
A point (c, f (c) on the graph of f where the concativity changes is called an inflection point, and c
is called an inflection value, given f (c) is defined.
3
4x3
− 4x2 − 12x + 2.
3
1. Determine where f is concave up, concave down.
Example 5. Given f (x) =
2. Find the inflection points.
4
The second derivative test for extrema: Suppose that f is defined on (a, b), f 0 (c) = 0, and c ∈
(a, b).
1. If f 00 (c) > 0, then f (c) is a relative minimum.
2. If f 00 (c) < 0, then f (c) is a relative maximum.
3. If f 00 (c) = 0, then the test is inconclusive.
x
.
Example 6. Use the second derivative test to find the relative extrema of f (x) = x x − x −
2
5
3
2