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There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
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Section 4.1Maximum and Minimum Values
V63.0121.021, Calculus I
New York University
November 9, 2010
Announcements
I Quiz 4 on Sections 3.3, 3.4, 3.5, and 3.7 next week (November 16,18, or 19)
Announcements
I Quiz 4 on Sections 3.3, 3.4,3.5, and 3.7 next week(November 16, 18, or 19)
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 2 / 34
Objectives
I Understand and be able toexplain the statement of theExtreme Value Theorem.
I Understand and be able toexplain the statement ofFermat’s Theorem.
I Use the Closed IntervalMethod to find the extremevalues of a function definedon a closed interval.
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 3 / 34
Notes
Notes
Notes
1
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 4 / 34
Optimize
Why go to the extremes?
I Rationally speaking, it isadvantageous to find theextreme values of a function(maximize profit, minimizecosts, etc.)
I Many laws of science arederived from minimizingprinciples.
I Maupertuis’ principle:“Action is minimizedthrough the wisdom ofGod.”
Pierre-Louis Maupertuis(1698–1759)V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 6 / 34
Notes
Notes
Notes
2
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Design
Image credit: Jason TrommV63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 7 / 34
Optics
Image credit: jacreativeV63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 9 / 34
Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 11 / 34
Notes
Notes
Notes
3
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Extreme points and values
Definition
Let f have domain D.
I The function f has an absolute maximum(or global maximum) (respectively,absolute minimum) at c if f (c) ≥ f (x)(respectively, f (c) ≤ f (x)) for all x in D
I The number f (c) is called the maximumvalue (respectively, minimum value) of fon D.
I An extremum is either a maximum or aminimum. An extreme value is either amaximum value or minimum value.
Image credit: Patrick QV63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 12 / 34
The Extreme Value Theorem
Theorem (The Extreme Value Theorem)
Let f be a function which is continuous on the closed interval [a, b]. Thenf attains an absolute maximum value f (c) and an absolute minimumvalue f (d) at numbers c and d in [a, b].
a bcmaximum
maximumvalue
f (c)
dminimum
minimumvalue
f (d)
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 13 / 34
No proof of EVT forthcoming
I This theorem is very hard to prove without using technical factsabout continuous functions and closed intervals.
I But we can show the importance of each of the hypotheses.
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 14 / 34
Notes
Notes
Notes
4
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Bad Example #1
Example
Consider the function
f (x) =
{x 0 ≤ x < 1
x − 2 1 ≤ x ≤ 2.|1
Then although values of f (x) get arbitrarily close to 1 and never biggerthan 1, 1 is not the maximum value of f on [0, 1] because it is neverachieved. This does not violate EVT because f is not continuous.
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 15 / 34
Bad Example #2
Example
Consider the function f (x) = x restricted to the interval [0, 1).
|1
There is still no maximum value (values get arbitrarily close to 1 but do notachieve it). This does not violate EVT because the domain is not closed.
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 16 / 34
Final Bad Example
Example
Consider the function f (x) =1
xis continuous on the closed interval [1,∞).
1
There is no minimum value (values get arbitrarily close to 0 but do notachieve it). This does not violate EVT because the domain is not bounded.
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 17 / 34
Notes
Notes
Notes
5
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 18 / 34
Local extrema
Definition
I A function f has a local maximum or relative maximum at c if f (c) ≥ f (x)when x is near c. This means that f (c) ≥ f (x) for all x in some open intervalcontaining c .
I Similarly, f has a local minimum at c if f (c) ≤ f (x) when x is near c .
|a
|blocal
maximumlocal
minimumglobalmax
local and globalmin
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 19 / 34
Local extrema
I So a local extremum must be inside the domain of f (not on the end).
I A global extremum that is inside the domain is a local extremum.
|a
|blocal
maximumlocal
minimumglobalmax
local and globalmin
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 20 / 34
Notes
Notes
Notes
6
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Fermat’s Theorem
Theorem (Fermat’s Theorem)
Suppose f has a local extremum at c and f is differentiable at c. Thenf ′(c) = 0.
|a
|blocal
maximumlocal
minimum
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 21 / 34
Sketch of proof of Fermat’s Theorem
Suppose that f has a local maximum at c .
I If x is slightly greater than c , f (x) ≤ f (c). This means
f (x)− f (c)
x − c≤ 0 =⇒ lim
x→c+
f (x)− f (c)
x − c≤ 0
I The same will be true on the other end: if x is slightly less than c,f (x) ≤ f (c). This means
f (x)− f (c)
x − c≥ 0 =⇒ lim
x→c−
f (x)− f (c)
x − c≥ 0
I Since the limit f ′(c) = limx→c
f (x)− f (c)
x − cexists, it must be 0.
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 22 / 34
Meet the Mathematician: Pierre de Fermat
I 1601–1665
I Lawyer and number theorist
I Proved many theorems,didn’t quite prove his lastone
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 23 / 34
Notes
Notes
Notes
7
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 25 / 34
Flowchart for placing extremaThanks to Fermat
Suppose f is a continuous function on the closed, bounded interval [a, b],and c is a global maximum point.
start
Is c anendpoint?
c = a orc = b
c is alocal max
Is fdiff’ble at
c?
f is notdiff at c
f ′(c) = 0
no
yes
no
yes
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 26 / 34
The Closed Interval Method
This means to find the maximum value of f on [a, b], we need to:
I Evaluate f at the endpoints a and b
I Evaluate f at the critical points or critical numbers x where eitherf ′(x) = 0 or f is not differentiable at x .
I The points with the largest function value are the global maximumpoints
I The points with the smallest or most negative function value are theglobal minimum points.
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 27 / 34
Notes
Notes
Notes
8
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 28 / 34
Extreme values of a linear function
Example
Find the extreme values of f (x) = 2x − 5 on [−1, 2].
Solution
Since f ′(x) = 2, which is never zero, we have no critical points and weneed only investigate the endpoints:
I f (−1) = 2(−1)− 5 = −7
I f (2) = 2(2)− 5 = −1
So
I The absolute minimum (point) is at −1; the minimum value is −7.
I The absolute maximum (point) is at 2; the maximum value is −1.
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 29 / 34
Extreme values of a quadratic function
Example
Find the extreme values of f (x) = x2 − 1 on [−1, 2].
Solution
We have f ′(x) = 2x, which is zero when x = 0. So our points to checkare:
I f (−1) = 0
I f (0) = − 1 (absolute min)
I f (2) = 3 (absolute max)
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 30 / 34
Notes
Notes
Notes
9
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Extreme values of a cubic function
Example
Find the extreme values of f (x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f ′(x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 andx = 1. The values to check are
I f (−1) = − 4 (global min)
I f (0) = 1 (local max)
I f (1) = 0 (local min)
I f (2) = 5 (global max)
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 31 / 34
Extreme values of an algebraic function
Example
Find the extreme values of f (x) = x2/3(x + 2) on [−1, 2].
Solution
Write f (x) = x5/3 + 2x2/3, then
f ′(x) =5
3x2/3 +
4
3x−1/3 =
1
3x−1/3(5x + 4)
Thus f ′(−4/5) = 0 and f is not differentiable at 0. So our points to checkare:
I f (−1) = 1
I f (−4/5) = 1.0341 (relative max)
I f (0) = 0 (absolute min)
I f (2) = 6.3496 (absolute max)
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 32 / 34
Extreme values of another algebraic function
Example
Find the extreme values of f (x) =√
4− x2 on [−2, 1].
Solution
We have f ′(x) = − x√4− x2
, which is zero when x = 0. (f is not
differentiable at ±2 as well.) So our points to check are:
I f (−2) = 0 (absolute min)
I f (0) = 2 (absolute max)
I f (1) =√
3
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 33 / 34
Notes
Notes
Notes
10
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010
Summary
I The Extreme Value Theorem: a continuous function on a closedinterval must achieve its max and min
I Fermat’s Theorem: local extrema are critical points
I The Closed Interval Method: an algorithm for finding global extrema
I Show your work unless you want to end up like Fermat!
V63.0121.021, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 9, 2010 34 / 34
Notes
Notes
Notes
11
Section 4.1 : Maximum and Minimum ValuesV63.0121.021, Calculus I November 9, 2010