3-1Extreme Values

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Chapter 3 Applications of DerivativesExtreme Values

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How many production runs to make??

You are planning this year’s printing schedule for your publishing company’s latest best-seller and your job is to minimize cost while ensuring that sales will be met!

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How many production runs to make??Here’s what you know…

Making predicts that it will sell 100,000 copies per month over the coming year. It costs $5,000 to set up the printing presses. Once the presses are set up, it costs $1 to print each book. It costs $0.01 to store each book for one month. You may make as many printing runs as you like, however to achieve efficient printing operations you must print the same number of books for each run and the runs must be evenly spaced throughout the year.C r 5,000r 1,200 ,000 72,000 r3

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Absolute Extrema(最值)A function f has an absolute (global) maximum at x = c if f ( x) f (c) for all x in the domain of f. A function f has a absolute (global) minimum at x = c if f ( x) f (c) for all x in the domain of f.Absolute Maximum

Absolute Minimum

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Extreme Value Theorem:If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.

Maximum & minimum at interior points

Maximum & minimum at endpoints

Maximum at interior point, minimum at endpoint

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Relative Extrema (极值)A function f has a relative (local) maximum (极大值) at x = c if f ( x) f (c) for all values of x close to c. A function f has a relative (local) minimum (极小值) at x = c if f ( x) f (c)for all values of x close to c.Relative Maximums

Relative Minimums6

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Absolute maximum (also local maximum) Local maximum

Local minimum

Local minimum Absolute minimum (also local minimum)

Local extremes are also called relative extremes.

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Absolute maximum (also local maximum) Local maximum

Local minimum

Notice that local extremes in the interior of the function occur where f is zero or f is undefined.

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Critical Points(临界点) of fA critical point of a function f is a point in the domain of f where1. f ( x) 0 (stationary point驻点)

2. f ( x) does not exist (singular point孤独点)

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Local Extreme Values: If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f exists at c, then f c 0 Note: Maximum and minimum points in the interior of a function always occur at critical points, but critical points are not always maximum or minimum values.

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Candidates for Relative Extrema1. Stationary points: x such that x is in the domain of f and f ( x) 0.2. Singular points: x such that x is in the domain of f and f ( x) is undefined. 3. Endpoints: endpoints of the domain (if any). Closed intervals have endpoints, open intervals do not.11

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Relative ExtremaEx. Find all the relative extrema off ( x) x3 6 x 2 1.

2 f ( x) 3x 12x 0

Stationary points: Endpoints: None

Singular points: None

x 0, 4

Relative max. f (0) = 14

Relative min. f (4) = –31

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Relative ExtremaEx. Find all the relative extrema off ( x) 3 x 3 3x . x 1 x2 1 Stationary points: f ( x) 2 3 x3 3x Singular points: x 0, 3Relative max. f (1) 3 2 Relative min. f ( 1) 3 2

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Increasing/Decreasing/Constant a, b , If f x 0 for each valueof x in an interval then f is increasingon a, b . a, b , If f x 0 for each valueof x in an interval then f is decreasingon a, b . a, b , If f x 0 for each valueof x in an interval then f is constanton a, b .14

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The First Derivative TestDetermine the sign of the derivative of f to the left and right of the critical point.left right

f(c) is a relative maximum f(c) is a relative minimum No relative extremum

No change

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The First Derivative TestEx. Find all the relative extrema of f ( x) x3 6 x 2 1.f ( x) 3x2 12x 0

3x( x 4) 0

Relative max. f (0) = 1

x 0, 4

Relative min. f (4) = -31 +

f

+0

4

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Absolute ExtremaIf a function f is continuous on a closed interval [a, b], then f attains an absolute maximum and minimum on [a, b]. Each extremum must occur at a critical point or an endpoint.

a

b

a

b

a

b

Attains absolute max. and min.

Attains absolute min. but not max.Interval open

No absolute min. or max.Not continuous17

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Example 1 Find the absolute extrema of f ( x) x 3x on , 4 . 2 2 f ( x) 3x 6x 3x( x 2)3 2

Critical values at x = 0, 2f (0) 0 f (2) 4Evaluate

Absolute Min.

7 1 f 8 2 f 4 16

Absolute Max.18

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EXAMPLE FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of f x x2/3 on the interval 2,3 .

f x x

2/31 3

2 f x x 3 2 f x 3 3 x

There are no values of x that will make the first derivative equal to zero. The first derivative is undefined at x=0, so (0,0) is a critical point. Because the function is defined over a closed interval, we also must check the endpoints.

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f x xAt:

2/3

D 2,3 actually a maximum or minimum, we try points on either side, without passing other critical points.

x 0 f 0 0 To determine if this critical point is

f 1 1

f 1 1

Since 0<1, this must be at least a local minimum, and possibly a global minimum.At:

x 2 f 2 2 1.5874 x 3

2 3

At:

f 3 3 2.08008

2 3

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f x xAt:

2/3

D 2,3 To determine if this critical point is actually a maximum or minimum, we Absolute try points on either side, without minimum: passing other critical point

s.

x 0 f 0 0

0, 0

f 1 1

f 1 1

Absolute maximum:

3, 2.08

Since 0<1, this must be at least a local minimum, and possibly a global minimum.2 3

At:

x 2 f 2 2 1.5874 x 3

At:

f 3 3 2.08008

2 3