Extreme Values of Functions

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Extreme Values of Functions

• Sec. 4.1 Notes

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The textbook gives the following example at the
start of chapter 4
Of course, this problem isnt entirely realistic,
since it is unlikely that you would have an
equation like this for your car.
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We could solve the problem graphically
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We could solve the problem graphically
On the TI-83, we use 2nd TRACE, 4 Maximum,
choose lower and upper bounds, and make a guess
and the calculator finds our answer.
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We could solve the problem graphically
On the TI-89, we use F5 (math), 4 Maximum,
choose lower and upper bounds, and the calculator
finds our answer.
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Notice that at the top of the curve, the
horizontal tangent has a slope of zero.
Traditionally, this fact has been used both as an
aid to graphing by hand and as a method to find
maximum (and minimum) values of functions.
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Even though the graphing calculator and the
computer have eliminated the need to routinely
use calculus to graph by hand and to find maximum
and minimum values of functions, we still study
the methods to increase our understanding of
functions and the mathematics involved.
Absolute extreme values are either maximum or
minimum points on a curve.
They are sometimes called global extremes.
They are also sometimes called absolute
extrema. (Extrema is the plural of the Latin
extremum.)
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Extreme values can be in the interior or the end
points of a function.
No Absolute Maximum
Absolute Minimum
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Absolute Maximum
Absolute Minimum
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Absolute Maximum
No Minimum
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No Maximum
No Minimum
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Extreme Value Theorem
If f is continuous over a closed interval, then
f has a maximum and minimum value over that
interval.
Maximum minimum at interior points
Maximum minimum at endpoints
Maximum at interior point, minimum at endpoint
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Local Extreme Values
A local maximum is the maximum value within some
open interval.
A local minimum is the minimum value within some
open interval.
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Absolute maximum
(also local maximum)
Local maximum
Local minimum
Local minimum
Absolute minimum
(also local minimum)
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Absolute maximum
(also local maximum)
Local maximum
Local minimum
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Local Extreme Values
If a function f has a local maximum value or a
local minimum value at an interior point c of its
domain, and if exists at c, then
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Critical Point
A point in the domain of a function f at
which or does not exist is a critical point
of f .
Note Maximum and minimum points in the interior
of a function always occur at critical points,
but critical points are not always maximum or
minimum values.
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There are no values of x that will make the first
derivative equal to zero.
The first derivative is undefined at x0, so
(0,0) is a critical point.
Because the function is defined over a closed
interval, we also must check the endpoints.
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To determine if this critical point is actually a
maximum or minimum, we try points on either side,
without passing other critical points.
Since 0lt1, this must be at least a local minimum,
and possibly a global minimum.
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To determine if this critical point is actually a
maximum or minimum, we try points on either side,
without passing other critical points.
Since 0lt1, this must be at least a local minimum,
and possibly a global minimum.
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Absolute maximum (3,2.08)
Absolute minimum (0,0)
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Finding Maximums and Minimums Analytically
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Critical points are not always extremes!
(not an extreme)
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(not an extreme)
p
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Find the absolute maximum and minimum of the
function
Find the critical numbers
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Find the absolute maximum and minimum of the
function
Check endpoints and critical numbers
The absolute maximum is 2 when x 2. The
absolute minimum is -13 when x -1.
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Find the absolute maximum and minimum of the
function
Find the critical numbers
NOTE We dont need to worry about x-3 in this
case because it doesnt lie within the interval
0,3.
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Find the absolute maximum and minimum of the
function
Check endpoints and critical numbers
The absolute maximum is 3 when x 0, 3 The
absolute minimum is 2 when x 1
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Find the absolute maximum and minimum of the
function
Find the critical numbers
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Find the absolute maximum and minimum of the
function
The absolute maximum is 1/4 when x ?/6,
5?/6 The absolute minimum is 2 when x 3?/2
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Find the absolute extrema of the given function
on the indicated interval.
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Find the absolute extrema of the given function
on the indicated interval.
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Find the extreme values and where they occur.
State whether they are global or local.
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Find the extreme values and where they occur.
State whether they are global or local.
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Find the extreme values and where they occur.
State whether they are global or local.
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Find the extreme values and where they occur.
State whether they are global or local.
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Homework