Increasing & Decreasing Functions and 1st Derivative Test

Lesson 4.3

Increasing/Decreasing Functions

• Consider the following function

• For all x < a we note that x1<x2 guarantees that f(x1) < f(x2)

f(x)

a

The function is said to be strictly increasing

The function is said to be strictly increasing

Increasing/Decreasing Functions

• Similarly -- For all x > a we note that x1<x2 guarantees that f(x1) > f(x2)

• If a function is either strictly decreasing or strictly increasing on an interval, it is said to be monotonic

f(x)

a

The function is said to be strictly

decreasing

The function is said to be strictly

decreasing

Test for Increasing and Decreasing Functions

• If a function is differentiable and f ’(x) > 0 for all x on an interval, then it is strictly increasing

• If a function is differentiable and f ’(x) < 0 for all x on an interval, then it is strictly decreasing

• Consider how to find the intervals where the derivative is either negative or positive

Test for Increasing and Decreasing Functions

• Finding intervals where the derivative is negative or positive Find f ’(x) Determine where

• Try for

• Where is f(x) strictly increasing / decreasing

• f ‘(x) = 0

• f ‘(x) > 0

• f ‘(x) < 0

• f ‘(x) does not exist

31( ) 9 2

3f x x x

Critical numbers

Critical numbers

Test for Increasing and Decreasing Functions

• Determine f ‘(x)

• Note graphof f’(x)

• Where is it pos, neg

• What does this tell us about f(x)f ‘(x) > 0 => f(x) increasing f ‘(x) > 0 => f(x) increasingf ‘(x) < 0 => f(x) decreasing

'( )f x

( )f x

First Derivative Test

• Given that f ‘(x) = 0 at x = 3, x = -2, and x = 5.25

• How could we find whether these points are relative max or min?

• Check f ‘(x) close to (left and right) the point in question

• Thus, relative min f ‘(x) < 0on left

f ‘(x) > 0on right

First Derivative Test

• Similarly, if f ‘(x) > 0 on left, f ‘(x) < 0 on right,

• We have a relative maximum

First Derivative Test

• What if they are positive on both sides of the point in question?

• This is called aninflection point

Examples

• Consider the following function

• Determine f ‘(x)

• Set f ‘(x) = 0, solve

• Find intervals

2 2( ) (2 1) ( 9)f x x x

Assignment A

• Lesson 4.3A

• Page 226

• Exercises 1 – 57 EOO

Application Problems

• Consider the concentrationof a medication in thebloodstream t hours afteringesting

• Use different methods to determine when the concentration is greatest Table Graph Calculus

3

3( ) 0

27

tC t t

t

Application Problems

• A particle is moving along a line and its position is given by

• What is the velocity of the particle at t = 1.5?

• When is the particle moving in positive/negative direction?

• When does the particle change direction?

2( ) 7 10s t t t

Application Problems

• Consider bankruptcies (in 1000's) since 1988

• Use calculator regression for a 4th degree polynomial Plot the data, plot the model Compare the maximum of the model, the

maximum of the data

1988 1989 1990 1991 1992 1993 1994

594.6 643.0 725.5 880.4 845.3 1042.1 835.2

Assignment B

• Lesson 4.3 B

• Page 227

• Exercises 95 – 101 all