MAT 1234 Calculus I Section 3.1 Maximum and Minimum Values
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Transcript of MAT 1234 Calculus I Section 3.1 Maximum and Minimum Values
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MAT 1234Calculus I
Section 3.1Maximum and Minimum
Values
http://myhome.spu.edu/lauw
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Next WebAssign 3.1 Quiz– 2.8
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1 Minute… You can learn all the important concepts
in 1 minute.
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1 Minute… High/low points – most of them are at points
with horizontal tangent
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1 Minute… High/low points – most of them are at points
with horizontal tangent.
Highest/lowest points – at points with horizontal tangent or endpoints
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1 Minute… You can learn all the important concepts
in 1 minute. We are going to develop the theory
carefully so that it works for all the functions that we are interested in.
There are a few definitions…
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Preview Definitions
• absolute max/min• local max/min• critical number
Theorems• Extreme Value Theorem• Fermat’s Theorem
The Closed Interval Method
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Max/Min We are interested in max/min values
• Minimize the production cost• Maximize the profit• Maximize the power output
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Definition (Absolute Max) has an absolute maximum at on if for all in ( =Domain of )
c
D
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Definition (Absolute Min) has an absolute maximum at on if for all in ( =Domain of )
cD
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Definition The absolute maximum and minimum
values of are called the extreme values of .
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x
Example 1y
Absolute max.
Absolute min.
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Definition (Local Max/Min) has an local maximum at if for all in some open interval containing .
has an local minimum at if for all in some open interval containing .
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x
Example 1y
Local max.
Local min.
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Q&A An end point is not a local max/min,
why?
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The Extreme Value Theorem If is continuous on a closed interval ,
then attains an absolute max value and an absolute min value at some numbers c and d in .
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The Extreme Value Theorem If is continuous on a closed interval ,
then attains an absolute max value and an absolute min value at some numbers c and d in .
No guarantee of absolute max/min if one of the 2 conditions are missing.
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Q&A Give 2 examples of functions on an
interval that do not have absolute max value.
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Example 2 (No abs. max/min) is not continuous on
𝑥
𝑦
𝑏𝑎
𝑦= 𝑓 (𝑥 )
𝑐
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Example 2 (No abs. max/min) The interval is not closed
𝑥
𝑦
𝑏𝑎
𝑦= 𝑓 (𝑥 )
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How to find Absolute Max./Min.? The Extreme Value Theorem guarantee
of absolute max/min if is continuous on a closed interval .
Next: How to find them?
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Fermat’s Theorem If has a local maximum or minimum at ,
and if exists, then
c x
y
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Q&A: T or F The converse of the theorem:If, then has a local maximum or minimum at .
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Definition (Critical Number) A critical number of a function is a
number c in the domain of such that either or does not exist.
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Critical Number (Translation) Critical numbers give all the potential
local max/min values
( ) 0 or f c DNE
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Critical Number (Translation) If the function is differentiable, critical
points are those such that
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Example 3Find the critical numbers of
3265)( xxxf
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Example 3Find the critical numbers of
3265)( xxxf
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The Closed Interval Method Idea: the absolute max/min values of a continuous function on a closed interval only occur at1. the local max/min (the critical numbers) 2. end points of the interval
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The Closed Interval Method To find the absolute max/min values of a continuous function on a closed interval :1. Find the values of at the critical numbers of in .2. Find the values of f at the end points.3. The largest of the values from steps 1 and 2 is the
absolute maximum value; the smallest of the those values from is the absolute minimum value.
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The Closed Interval Method To find the absolute max/min values of a continuous function on a closed interval :1. Find the values of at the critical numbers of in .2. Find the values of at the end points.3. The largest of the values from steps 1 and 2 is the
absolute maximum value; the smallest of the those values from is the absolute minimum value.
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The Closed Interval Method To find the absolute max/min values of a continuous function on a closed interval :1. Find the values of at the critical numbers of in .2. Find the values of at the end points.3. The largest of the values from steps 1 and 2 is the
absolute maximum value; the smallest of the those values from is the absolute minimum value.
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Example 4Find the absolute max/min values of
]5,3[on 112)( 3 xxxf
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Expectations: Formal Conclusion