maximum and minimum.pptx

8/14/2019 Maximum and Minimum.pptx

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Maximum and Minimum Values

Prepared by: Miss Rose Anne B. Camacho


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An important application of the

derivative is to determine where a function attains its maximum andminimum (extreme) values.


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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 openinterval containing c , on which f is defined,such that

for all x in the interval

f(c) f(x)


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Figures 1 and 2 show a portion of the

graph of a function having a relativemaximum value at c.

a bc

Figure 1 Figure 2

a bc


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Definition of a Relative Minimum Value

The function f has a relative minimum valueat the number c if there exists an openinterval containing c , on which f is defined,such that

for all x in the interval

f(c) f(x)


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Figures 3 and 4 show a portion of the

graph of a function having a relativemaximum value at c.

a bc

Figure 3 Figure 4

a bc


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ILLUSTRATION 1:

Let f be the function defined by f(x) = x2 4x + 5

f '(x) = 2x 4because f(2) = 0 , f may have a relativeextremum at 2. Because f(2) = 1 and1 < f(x) when either x < 2 or x > 2 . Definitionguarantees that f has a relative minimum valueat 2 .


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NOTE that

f(c) can be equal to zero even f does

not have a relative extremum at c.


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ILLUSTRATION 2:

Consider the function f defined by f(x) = (x 1) 3 + 2

f '(x) = 3(x 1) 2

because f(1) = 0 , f may have a relativeextremum at 1. However f(1) = 2 and

2 > f(x) when either x < 1 and 2 < f(x) whenx >1 . Neither definition for maximum orMinimum value applies, f does not have a

relative extremum at 1.


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NOTE that

A function may have a relative

extremum at a number at which thederivative fails to exist.


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ILLUSTRATION 3:

Let f be the function defined by

f(3) = 2 and f(3) = -1 f(3) = dne

because f(3) = 5 and5 > f(x) when either x < 3 and 3 < x . f has a relative maximum at 3.

x3 if 8

3xif 12

x

x x f


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ILLUSTRATION 3:


f(3) = 2 and f(3) = -1 f(3) = dne

because f(3) = 5 and5 > f(x) when either x < 3 and 3 < x . f has a relative maximum at 3.

x3 if 8

3xif 12

x

x x f


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NOTE that

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 havea relative extremum there.


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ILLUSTRATION 4:


Furthermore, f(0) does not exist. The function has no relative extrema .

31

x x f

The domain of f is the set of all real numbers 0x if

3

1'

3

2

x x f


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In summary.

If a function f is a defined at a number

c, a necessary condition for f to have arelative extremum there is that either f(c) = 0 or f(c) does not exist. But

this condition is not sufficient.


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The Calculus 7 by Louis Leithold, pp. 210 212

Reference:

maximum and minimum.pptx

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