Relative Extrema

Lesson 5.5

Video Profits Revisited

Recall our Digitari manufacturer

Cost and revenue functions• C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250• R(x) = 8.4x - .002x2 0 ≤ x ≤ 2250

Cost, revenue, and profit functions

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Video Profits Revisited

Digitari wants to know how many to make and sell for maximum profit

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• Profits increasing on this interval

• Slope > 0

• Profits increasing on this interval

• Slope > 0

• Profits decreasing on this interval

• Slope < 0

• Profits decreasing on this interval

• Slope < 0

Maximum profit when

• Profits neither increasing nor decreasing

• Slope = 0

Maximum profit when

• Profits neither increasing nor decreasing

• Slope = 0

Relative Maximum

Given f(x) on open interval (a, b) with point c in the interval

Then f(c) is the relative maxif f(x) ≤ f(c) for all x in (a, b)

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a b( )

c

Relative Minimum

Given f(x) on open interval (a, b) with point c in the interval

Then f(c) is the relative minif f(x) ≥ f(c) for all x in (a, b)

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ab

( )c

Relative Max, Min

Note• Relative max or min does

not guarantee f '(x) = 0

Important Rule:• If a function has a

relative extremum at c• Then either c a critical number

or c is an endpoint of the domain

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First Derivative Test

Given • f(x) differentiable on (a, b), except possibly

at c• c is only critical

number in interval

f(c) is relative max if• f '(x) > 0 on (a, c) and• f '(x) < 0 on (c, b)

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a b( )

c

First Derivative Test

Given • f(x) differentiable on (a, b), except possibly

at c• c is only critical

number in interval

f(c) is relative min if• f '(x) < 0 on (a, c) and• f '(x) > 0 on (c, b)

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ab

( )c

First Derivative Test

Note two other possibilities• f '(x) < 0 on both sides

of critical point

• f '(x) > 0 on both sidesof critical point

Then no relative extrema

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Finding Relative Extrema

Strategy

Find critical points

Check f '(x) on either side• Negative on left, positive on right → min• Positive on left, negative on right → max

Try it!

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2( ) 4 6f x x x 2/32 3y x x 2 6 9

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x xy

x

Application

Back to Digitari … cost and revenue functions• C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250• R(x) = 8.4x - .002x2 0 ≤ x ≤ 2250

Just what is that number of units to market for maximum profit?

What is the maximum profit?

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Assignment

Lesson 5.2

Page 327

Exercises 1 – 53 EOO

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