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Glossary

A

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][-2, 2], the function f(x)=x2f(x) = x^2 has an absolute minimum at x=0x=0 and an absolute maximum at x=±2x=\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)=−x2+5f(x) = -x^2 + 5 on [−1,1][-1, 1], the absolute maximum is f(0)=5f(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)=x2−3f(x) = x^2 - 3 on [−2,2][-2, 2], the absolute minimum is f(0)=−3f(0)=-3.

C

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)=x3−3xf(x) = x^3 - 3x on [0,2][0, 2], you would apply the Candidates Test by comparing f(0)f(0), f(2)f(2), and f(x)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)f(x) = \sin(x) on [0,2π][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)=x2+4xf(x) = x^2 + 4x, setting its derivative f′(x)=2x+4f'(x) = 2x + 4 to zero gives x=−2x = -2, which is a critical point.

F

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)=x3−6x2f(x) = x^3 - 6x^2 has critical points, you would calculate its first derivative, f′(x)=3x2−12xf'(x) = 3x^2 - 12x, and set it to zero.

R

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)=x3−3xf(x) = x^3 - 3x, the point (1,−2)(1, -2) is a relative minimum because the function decreases before it and increases after it in its immediate vicinity.