Lecture#20 Analysis of Function II.pptx

MTH 101
Calculus 1
ANALYSIS OF FUNCTIONS - II:
EXTREMA OF THE FUNCTION
Zertaisha Nasir
Mathematics Department,
COMSATS University Islamabad, Wah Campus.
APPLICATIONS OF DERIVATIVES
Lecture# 20

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REVIEW OF LAST LECTURE:
• Increasing and decreasing functions
• Concavity
• Inflection points

TOPICS TO BE COVERED:
• Extreme points
• Relative/ local extreme points
• Critical and stationary points
• First derivative test
• Second derivative test
• Absolute extreme points
• Roll’s theorem
• Mean value theorem

OBJECTIVES:
• To be able to find the relative/ local extreme points and absolute
extreme points of the function.
• To learn the use of Roll’s theorem and Mean value theorem.

EXTREME POINTS:
If f has either maximum or minimum at , then f is said to have a extremum at .
• Relative/ local extremum
• Absolute extremum

RELATIVE/ LOCAL EXTREMA:
If f has either a relative maximum or a relative minimum at , then f is said to have a
relative extremum at .
A function f is said to have a relative maximum at if there is an open interval containing x0
on which f() is the largest value, that is, f() ≥ f(x) for all x in the interval.
RELATIVE MAXIMUM:
f is said to have a relative minimum at if there is an open interval containing on which f() is
the smallest value, that is, f() ≤ f(x) for all x in the interval.
RELATIVE MINIMUM:

• f(x) = has a relative
minimum at x = 0 but
no relative maxima.
• f(x) = has no
relative extrema.
• f(x) = − 3x + 3 has a relative
maximum at x = −1 and a
relative minimum at x = 1.
Example 1:

• has relative minima at x = −1 and x = 2
and a relative maximum at x = 1.
• f(x) = cos x has relative maxima at all
even multiples of π and relative
minima at all odd multiples of π.

We define a critical point for a function f to be a
point in the domain of f at which either the graph of f
has a horizontal tangent line or f is not differentiable.
To distinguish between the two types of critical
points we call x a stationary point of f if .
CRITICAL POINT AND STATIONARY POINT:

Example 2: Find all critical points of − 3x + 1.
Solution: − 3x + 1
− 3
𝑓 ′
(𝑥 )=0 ⇒3 𝑥2
− 3=0
⇒ 𝑥2
− 1=0
⇒ ( 𝑥 −1) (𝑥 +1)=0
⇒ 𝑥=± 1

Example 3: Find all critical points of .
Solution:
Thus x = 0 and x = 2 are critical points and x = 2
is a stationary point.

A function f has a relative extremum at those critical points where changes sign.

Example 4: Find relative extrema of − 3x + 1.
Solution: − 3x + 1
− 3
𝑓 ′
(𝑥 )=0 ⇒3 𝑥2
− 3=0
⇒ 𝑥2
− 1=0
⇒ ( 𝑥−1) (𝑥+1)=0

Example 5:
Solution:
Find the relative extrema of .
Critical points are
𝑓 ′ ′
(0 )=0
𝑓 ′ ′
(1)=30
𝑓 ′ ′
(−1)=−30
f has inflection point at
f has relative minimum at .
f has relative maximum at .

ABSOLUTE EXTREMA:
f has an absolute maximum at if f(x) ≤ f() for all x in the interval.
ABSOLUTE MAXIMUM:
Consider an interval in the domain of a function f and a point x0 in that interval. f has
an absolute extremum at if it has either an absolute maximum or an absolute
minimum at that point.
ABSOLUTE MINIMUM:
f has an absolute minimum at if f() ≤ f(x) for all x in the interval.

Lecture#20 Analysis of Function II.pptx

Example 6:
Solution:
Find the absolute maximum and minimum values of the function on the
interval [1, 5], and determine where these values occur.
𝑓 (𝑥)=2𝑥3
−15 𝑥2
+36𝑥
𝑓 ′ (𝑥)=6 𝑥2
−30𝑥+36
𝑓 ′ ( 𝑥)=0 ⇒ 6 𝑥2
−30 𝑥+36=0
⇒ 𝑥2
−5 𝑥+6=0
⇒ ( 𝑥 −2) ( 𝑥 −3 )=0
⇒ 𝑥=2,3
Critical
points
Absolute extrema can occur at .
End
points
• the absolute maximum of f on [1, 5] is
55, occurring at x = 5.
• the absolute minimum of f on [1, 5] is
23, occurring at x = 1.

is a point in the interval (1, 4) at which
Example 7:
Solution:

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Lecture#20 Analysis of Function II.pptx

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