Download Math 163 Notes Section 5.3

Section 5.3 Higher Derivatives, Concavity, and the Second Derivative Test
The second derivative of a function f, written f " , gives the rate of change of
the derivative of f.
Given f  x   x 4  2 x 3  3 x 2  5 x  7 , then
f ' x  4x3  6x2  6x  5
First derivative of f
f "  x   12 x 2  12 x  6
Second derivative of f
f "'  x   24 x  12
Third derivative of f
f ( 4 )  x   24
Fourth derivative of f
The first derivative of a function represents the rate of change of the function.
The second derivative, then, represents the rate of change of the first
derivative—it indicates how fast the function is increasing or decreasing.
 If st  represents position at time t, then
o vt   s' t  gives velocity at time t
o at   v' t   s" t  represents acceleration at time t
5.3 - 2
Concavity of a Graph
A function is concave upward on an interval a, b  if the graph of the function
lies above its tangent line at each point of a, b  . A function is concave
downward on a, b  if the graph of the function lies below its tangent line at
each point of a, b  . A point where a graph changes concavity is called an
inflection point.
 To find an inflection point, first find the second derivative. Set the
second derivative equal to 0 and solve for x. Now evaluate the original
function at the x-value to determine the y-coordinate of the inflection
point.
 The x-coordinate of the inflection point(s) divides a number line into
intervals. Choose a test point in each interval to determine concavity. (If
no points of inflection, divide number line where f " is undefined.)
o If f " is positive – concave up
o If f " is negative – concave down
o If f " has no inflection points and is never undefined, the function
does not change concavity. Choose any test point to determine if
the function is at all points concave up or concave down.
5.3 - 3
Second Derivative Test:
 Find critical numbers (where f ' ( x )  0 and where f ' ( x ) is undefined.)
 Find points of inflection (where f " ( x )  0 and where f " ( x ) is
undefined.)
 Plug critical numbers into f " ( x )
o If f " (c )  0  concave down and relative maximum
o If f " (c )  0  concave up and relative minimum
o If f " (c )  0  second derivative test fails and you must use the
first derivative test
Remember: Always plug all critical numbers back into the original function
to find the y-value that corresponds to it.
Work #1 - 8
The law of diminishing returns in economics is related to the idea of concavity.
When a graph is concave up, the rate of change of the slope is increasing.
However, when the graph is concave down, the rate of change of the slope is
decreasing. Thus, further input beyond c dollars produces diminishing
returns. The inflection point at c, f (c ) is called the point of diminishing
returns.
Work #9 - 11