(DG1) Graphically Finding Intervals of Increase / Decrease#

By the end of the lesson you will be able to:

  • locate the intervals of increase and the intervals of decrease given the graph of a function.

  • locate the local maximums and local minimums given the graph of a function.

Lecture Videos#

Increasing / Decreasing#

Extreme Values#

Concavity#

Example 2#

First Derivative Rule#

Second Derivative Rule#

Examples 5 and 6#

Example 7#

Increasing / Decreasing#

Increase / Decrease

Increasing on an Interval: the graph continuously rises as \(x\) goes from left to right through the interval.

\[ x_1<x_2 \implies f(x_1)<f(x_2) \]

Decreasing on an Interval: the graph continuously falls as \(x\) goes from left to right through the interval.

\[ x_1<x_2 \implies f(x_1)>f(x_2) \]

Example 1#

Determine if the following graphs are increasing, decreasing, or neither at \(x=c\).

Increasing
Flat
Decreasing

Relative Maximum and Minimum#

Relative Maximum / Minimum

A local maximum value of \(f\) is a number \(f(c)\) such that \(f(c)\geq f(x)\) for \(x\) near \(c\).

A local minimum value of \(f\) is a number \(f(c)\) such that \(f(c)\leq f(x)\) for \(x\) near \(c\).

Maximum
Minimum

Example 2#

Use the graph of \(y=f(x)\) given below to find:

  • all intervals of increase and all intervals of decrease

  • the \(x\)-values of all local maximums and local minimums

Intervals

Example 3#

Use the graph of \(y=f(x)\) given below to find:

  • all intervals of increase and all intervals of decrease

  • the \(x\)-values of all local maximums and local minimums

Intervals

Critical Number

The number \(x=c\) is a critical number of \(f\) provided either:

\[ f'(c)=0 \qquad \text{or} \qquad f'(c) \text{ DNE} \]

and \(c\) is in the domain of \(f\).