behavior of functions and their graphs

Behavior of Functions and

Their Graphs,

ROMMEL O. GREGORIO

OUTLINE

Maximum and minimum values

Increasing and decreasing functions and the first-

derivative test

Concavity, points of inflection, and the second-

derivative test

Sketching graphs of functions and their

derivatives

Summary of sketching graphs of functions

Optimization Problems

MAXIMUM AND MINIMUM FUNCTION

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Definition of a Relative Maximum Value

The function f has a relative maximum value at

the number c if there exists an open interval

containing c, on which f is defined, such that

f(c)≥f(x) for all x in the interval.

Definition of a Relative Minimum Value

The function f has a relative minimum value at

the number c if there exists an open interval

containing c, on which f is defined, such that

f(c)≤f(x) for all x in the interval.


VALUES Definition of an Absolute Maximum Value

The function f has an absolute maximum value on an

interval if there is some number c in the interval such

that f(c)≥f(x) for all x in the interval. The number f(c)

is then the absolute maximum value of f on the

interval.

Definition of an Absolute Minimum Value

The function f has an absolute minimum value on an

interval if there is some number c in the interval such

that f(c)≤f(x) for all x in the interval. The number f(c)

is then the absolute minimum value of f on the

interval.


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Theorem

If f(x) exists for all values of x in the open interval

(a, b), and if f has a relative extremum at c,

where a<c<b, and if f’(c) exists, then f’(c)=0

If f has a relative extremum at c, and if f’(c)

exists, then the graph of f must have a horizontal

tangent line at the point where x=c.


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Illustration

f(x)=x2-4x+5

x

y


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QUESTION: If f’(c)=0, does

it mean that we have a

relative extremum at c?


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Consider

f(x)=(x-1)3+2

x

y


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QUESTION: If f’(c) does not

exist, is it possible for f to

have a relative extremum

at c?


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Consider

𝑓 𝑥 = 2𝑥 − 1 𝑖𝑓 𝑥 ≤ 38 − 𝑥 𝑖𝑓 𝑥 > 3

x

y


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Remarks:

1. f’(c) can be equal to zero even if f does not have

a relative extremum at c.

2. f may have a relative extremum at a number at

which the derivative fails to exist.

3. it is possible that a function f can be defined at a

number c where f’(c) does not exist and yet f may

not have a relative extremum there.


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In summary, if a function f is defined

at a number c, a necessary condition

for f to have a relative extremum there

is that either f’(c)=0 or f’(c) does not

exist. But this condition is not

sufficient.


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Definition of a Critical Number

If c is a number on the domain of the function f,

and if either f’(c)=0 or f’(c) does not exist, then c

is a critical number of f.

Examples

1. f x = x4 + 4x3 − 2x2 − 12x

2. f x =x2+4

x−2

3. f x = x − 2


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Example 1

f(x)=4-3x on (-1, 4]

x

y


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Example 2

𝑓 𝑥 = 4 − 𝑥2; (-2, 2)

x

y


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Example 3

𝑔 𝑥 = 𝑥 + 1 𝑖𝑓 𝑥 ≠ −13 𝑖𝑓 𝑥 = −1

; [-2, 1]

x

y


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Example 4

ℎ 𝑥 =4

(𝑥−3)2; [2, 5)

x

y


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The Extreme Value Theorem

If the function f is continuous on

the closed interval [a, b], then f

has an absolute maximum value

and an absolute minimum value on

[a, b].


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Steps in Finding the absolute Extremum of f on [a,

b]

1. Find the function values at the critical numbers of

f on (a, b)

2. Find the values of f(a) and f(b)

3. The largest of the values from steps 1 and 2 is the

absolute maximum value, and the smallest of the

values is the absolute minimum value


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Examples. Find the absolute extrema of the

following functions on the given interval.

1. f x = x3 + 5x − 4, [-3, -1]

2. g x =x

x+2, [-1, 2]

3. h x = x4 − 8x2 + 16, [-1, 2]

4. F x = x3 − 5x − 4, [-3, -1]