Increasing and Decreasing FunctionsLesson 5.1

The Ups and DownsThink of a function as a roller coaster going from left to right

Uphill• Slope > 0• Increasing

function

Downhill • Slope < 0• Decreasing function

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Definitions

Given function f defined on an interval• For any two numbers x1 and x2 on the interval

Increasing function • f(x1) < f(x2) when x1 < x2

Decreasing function• f(x1) > f(x2) when x1< x2

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X2X1 X1X2

f(x)

Increasing/Decreasing and the Derivative

Assuming existence of derivative on interval

If f '(x) > 0 for each x• f(x) increasing on

interval

If f '(x) < 0 for each x• f(x) decreasing on

interval 4

What if f '(x) = 0 on the interval? What could you say about f(x)?

What if f '(x) = 0 on the interval? What could you say about f(x)?

Check These Functions

By graphing on calculator, determine the intervals where these functions are • Increasing• Decreasing

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3 22( ) 4 2

3f x x x x

2 5y x 3

( )4

xf x

x

Critical Numbers

Definition

Numbers c in the domain of f where• f '(c) = 0

• f '(c) does not exist

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Critical PointsCritical Points2

( )2

xf x

x

2

( )2

xf x

x

Applying Derivative Test

Given a function f(x)

Determine the derivative f '(x)

Find critical points … • Where f '(x) = 0 or f '(x) does not exist

Evaluate derivative between or on either side of the critical points

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3 22( ) 4 2

3f x x x x Try it with this function

Applications

Digitari, the great video game manufacturer determines its cost and revenue functions• C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250• R(x) = 8.4x - .002x2 0 ≤ x ≤ 2250

Determine the interval(s) on which the profit function is increasing

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Assignment

Lesson 5.1

Page 313

Exercises 1 – 57 EOO

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