Section 4.3Increasing and Decreasing Functions and

the First Derivative Test

Relative Maximum f(x)

Relative Extrema

Relative Minimum

f(x)

Relative Extrema

f(x)

Relative Extrema

f(x)

Relative Extrema

Increasing and Decreasing

𝑓 ’ (𝑥 )<0• is

decreasing

𝑓 ’ (𝑥 )=0• is

constant

𝑓 ’ (𝑥 )>0• is

increasing

Use the graph to find the (a) largest open interval where the function is decreasing and the (b) largest open interval where its increasing.

Example 1

Use the graph to estimate where the fnc. is increasing or decreasing. Then find the open intervals analytically.

Example 2

𝒇 (𝒙 )=𝒙𝟒−𝟐 𝒙𝟐

Identify the intervals where the function is increasing or decreasing.

Example 3

𝑓 (𝑥 )=27 𝑥−𝑥3

Identify the intervals where the function is increasing or decreasing.

Example 4

𝑓 (𝑥 )=𝑥+4𝑥

Identify the intervals where the function is increasing or decreasing and locate all relative extrema.

Example 5

𝑓 (𝑥 )=𝑥4−32𝑥+4

Identify the intervals where the function is increasing or decreasing and locate all relative extrema.

Example 6

𝑓 (𝑥 )=𝑥+4𝑥2

Identify the intervals where the function is increasing or decreasing and locate all relative extrema.

Example 7

𝑓 (𝑥 )=(𝑥−1)𝑒𝑥

Identify the intervals where the function is increasing or decreasing and locate all relative extrema.

Example 8

𝑓 (𝑥 )=𝑥+2sin 𝑥 ,0<𝑥<2𝜋

The graph of is given. Sketch the graph of .Example 9

The graph of is given. Sketch the graph of .Example 10

Be sure to be practicing the given problem sets!

Questions?