Glossary

Absolute (Global) Extrema

Criticality: 3

The overall maximum and minimum values of a function over its entire domain or a specified closed interval.

Example:

On the interval $[-2, 2]$ , the function $f(x) = x^2$ has an absolute minimum at $x=0$ and an absolute maximum at $x=\pm 2$ .

Absolute Maximum

Criticality: 3

The largest y-value a function attains over its entire domain or a specified closed interval.

Example:

For $f(x) = -x^2 + 5$ on $[-1, 1]$ , the absolute maximum is $f(0)=5$ .

Absolute Minimum

Criticality: 3

The smallest y-value a function attains over its entire domain or a specified closed interval.

Example:

For $f(x) = x^2 - 3$ on $[-2, 2]$ , the absolute minimum is $f(0)=-3$ .

Candidates Test

Criticality: 3

A systematic method used to find the absolute extrema of a continuous function on a closed interval by evaluating the function at its critical points and the interval's endpoints.

Example:

To find the absolute maximum of $f(x) = x^3 - 3x$ on $[0, 2]$ , you would apply the Candidates Test by comparing $f(0)$ , $f(2)$ , and $f(x)$ at any critical points within the interval.

Closed Interval

Criticality: 2

An interval that includes its endpoints, typically denoted using square brackets, e.g., $[a, b]$.

Example:

When finding the absolute extrema of $f(x) = \sin(x)$ on $[0, 2\pi]$ , the domain is a closed interval.

Critical Point

Criticality: 3

A point in the domain of a function where its first derivative is either zero or undefined. These points are potential locations for relative or absolute extrema.

Example:

For $f(x) = x^2 + 4x$ , setting its derivative $f'(x) = 2x + 4$ to zero gives $x = -2$ , which is a critical point.

First Derivative

Criticality: 3

The derivative of a function, which represents the instantaneous rate of change of the function and is used to find critical points and determine intervals of increasing or decreasing behavior.

Example:

To find where $f(x) = x^3 - 6x^2$ has critical points, you would calculate its first derivative, $f'(x) = 3x^2 - 12x$ , and set it to zero.

Relative/Local Extrema

Criticality: 2

The maximum or minimum value of a function within a specific open interval, where the function changes from increasing to decreasing (maximum) or vice versa (minimum).

Example:

For the function $f(x) = x^3 - 3x$ , the point $(1, -2)$ is a relative minimum because the function decreases before it and increases after it in its immediate vicinity.

Glossary

Absolute (Global) Extrema

Criticality: 3

The overall maximum and minimum values of a function over its entire domain or a specified closed interval.

Example:

On the interval $[-2, 2]$ , the function $f(x) = x^2$ has an absolute minimum at $x=0$ and an absolute maximum at $x=\pm 2$ .

Absolute Maximum

Criticality: 3

The largest y-value a function attains over its entire domain or a specified closed interval.

Example:

For $f(x) = -x^2 + 5$ on $[-1, 1]$ , the absolute maximum is $f(0)=5$ .

Absolute Minimum

Criticality: 3

The smallest y-value a function attains over its entire domain or a specified closed interval.

Example:

For $f(x) = x^2 - 3$ on $[-2, 2]$ , the absolute minimum is $f(0)=-3$ .

Candidates Test

Criticality: 3

A systematic method used to find the absolute extrema of a continuous function on a closed interval by evaluating the function at its critical points and the interval's endpoints.

Example:

To find the absolute maximum of $f(x) = x^3 - 3x$ on $[0, 2]$ , you would apply the Candidates Test by comparing $f(0)$ , $f(2)$ , and $f(x)$ at any critical points within the interval.

Closed Interval

Criticality: 2

An interval that includes its endpoints, typically denoted using square brackets, e.g., $[a, b]$.

Example:

When finding the absolute extrema of $f(x) = \sin(x)$ on $[0, 2\pi]$ , the domain is a closed interval.

Critical Point

Criticality: 3

A point in the domain of a function where its first derivative is either zero or undefined. These points are potential locations for relative or absolute extrema.

Example:

For $f(x) = x^2 + 4x$ , setting its derivative $f'(x) = 2x + 4$ to zero gives $x = -2$ , which is a critical point.

First Derivative

Criticality: 3

The derivative of a function, which represents the instantaneous rate of change of the function and is used to find critical points and determine intervals of increasing or decreasing behavior.

Example:

To find where $f(x) = x^3 - 6x^2$ has critical points, you would calculate its first derivative, $f'(x) = 3x^2 - 12x$ , and set it to zero.

Relative/Local Extrema

Criticality: 2

The maximum or minimum value of a function within a specific open interval, where the function changes from increasing to decreasing (maximum) or vice versa (minimum).

Example:

For the function $f(x) = x^3 - 3x$ , the point $(1, -2)$ is a relative minimum because the function decreases before it and increases after it in its immediate vicinity.