Transcript Business Calculus - Front Range Community College

Business Calculus
Extrema
 Extrema: Basic Facts
Two facts about the graph of a function will help us in
seeing where extrema may occur.
1. The intervals where the graph is rising or falling.
2. The possible shapes of the graph at its highest or
lowest points.
 2.1 Intervals of Increase and Decrease:
 A function is increasing if the function is rising from
left to right.
A function is decreasing if the function is falling from
left to right.
 A critical value of f is a point of the graph where f might
change from increasing to decreasing or decreasing to
increasing.
 Critical values are found in two ways:
f has a critical value at x = c if either of the following are true:
1.
f (c)  0
2.
f (c) dne
 Intervals of increase and intervals of decrease:
We can use derivatives to determine where a function is
increasing and where it is decreasing.
1. If f ′(x) > 0 for all x in an interval, then f is increasing
on that interval.
2. If f ′(x) < 0 for all x in an interval, then f is decreasing
on that interval.
 2.2 Concavity:
 A function is concave up if the function is in the shape of an
upright bowl.
A function is concave down if the function is in the shape of an
upside down bowl.
On a given interval, the second derivative will tell us whether
the function is concave up or concave down.
1. If f ʺ(x) > 0 for all x in an interval, then f is concave up
on that interval.
2. If f ʺ(x) < 0 for all x in an interval, then f is concave down
on that interval.
 2.1 Relative extrema:
A function f has a relative maximum at x = c if
1. c is in the domain of f and
2. f (c) is greater than all other y values of the function in an
interval containing c.
A function f has a relative minimum at x = c if
1. c is in the domain of f and
2. f (c) is less than all other y values of the function in an
interval containing c.
 2.4 Absolute extrema:
A function f has an absolute maximum at x = c if
1. c is in the domain of f and
2. f (c) is greater than all other y values of the function
on its entire domain.
A function f has an absolute minimum at x = c if
1. c is in the domain of f and
2. f (c) is less than all other y values of the function
on its entire domain.
Important Fact: relative and absolute extrema can occur only
at critical values or endpoints of the function.
English translation:
“What is the relative (or absolute) extrema” means give a y value.
“Where does the relative (or absolute) extrema occur” means
give an x value.
 To find relative and absolute extrema of a function:
A) Find all critical points and endpoints. This is the list
of possible extrema.
B) Test each critical point and endpoint to determine if it
is a minimum, maximum, or neither one.
 Testing Critical Values for Relative Extrema
There are two ways to test critical points for relative extrema.
2.1 First Derivative Test:
Note:
A function has a relative minimum at a point in its domain
if it changes from decreasing to increasing at that point.
A function has a relative maximum at a point in its domain
if it changes from increasing to decreasing at that point.
Because the derivative can tell us about the direction of the
curve, we can use the derivative to test critical values to determine
if it gives a relative minimum, relative maximum, or neither one.
 First Derivative Test
f has a relative minimum at the critical point x = c if
1. c is in the domain of f and
2. f ′(x) < 0 for x on the left of c and f ′(x) > 0 for x on the right of c.
f has a relative maximum at the critical point x = c if
1. c is in the domain of f and
2. f ′(x) > 0 for x on the left of c and f ′(x) < 0 for x on the right of c.
We will use a number line to organize this information and
find relative extrema at critical points and endpoints.
2.2 Second Derivative Test:
Note:
If a critical value comes from setting y′ = 0, then it sits at either
the top of an upside down bowl (concave down),
or the bottom of an upright bowl (concave up).
We can use this information to determine if a critical value is a
relative minimum or relative maximum, but only if the critical
value comes from setting y′ = 0.
 Second Derivative Test
f has a relative minimum at the critical point x = c if
1. c is in the domain of f found by f ′(x) = 0 and
2. f ʺ(c) > 0
f has a relative maximum at the critical point x = c if
1. c is in the domain of f found by f ′(x) = 0 and
2. f ʺ(c) < 0
It is important to note that we are plugging c into the second
derivative to test whether x = c will give a minimum or maximum.
 Testing Critical Values for Absolute Extrema
There are two ways to test critical values and endpoints for
absolute extrema, the extreme value theorem and a special case
of the second derivative test.
 Extreme Value Theorem (Principal 1)
If f is continuous on a closed interval [a, b], then f will attain
its absolute minimum and its absolute maximum.
This means that if we make a list of all critical values and
endpoints (x values), and then find each corresponding y value,
the highest y value is the absolute maximum and the lowest y
value is the absolute minimum.
 Using the 2nd derivative test for absolute extrema
If we are interested in a continuous function, and it has only one
critical value which is found by solving y′ = 0, then we can use
the second derivative test to determine absolute extrema at that
critical value.
Note: it is not required that we are working on a closed interval.
Since there is only one critical value, this value will give either an
absolute minimum or an absolute maximum.
 2nd derivative test (Principal 2)
For a single critical value of x = c which is in the domain of f
and comes from y′ = 0
1. f will have an absolute minimum at x = c if f ʺ(c) > 0.
2. f will have an absolute minimum at x = c if f ʺ(c) > 0.
Relative Extrema
Find all critical values and endpoints.
If one or more
critical values come
from y′ dne, use the
first derivative test:
draw the first
derivative number
line.
or
For any critical
values coming from
y′=0, we can use
the second
derivative test:
plug the CVs into
the second
derivative.
Absolute Extrema
Find all critical values and
endpoints. Make a list of these
values.
If f is continuous on
a closed interval,
test these values
using the extreme
value theorem.
If there is only one
critical value coming
from y′=0, and no
usable end points,
use the second
derivative test.