How to Determine Extrema using Candidates Test

Candidates Test

A method used to determine the absolute extrema (maximum and minimum) of a function on a closed interval.

Closed Interval

An interval [a,b] that includes its endpoints a and b.

Continuous Function

A function f(x) is continuous on an interval if there are no breaks, jumps, or holes in its graph.

Critical Points

Points where the derivative f′(x)=0 or where the derivative does not exist.

Derivative

A measure of how a function changes as its input changes, represented by f′(x).

Endpoints

The values at the boundaries of the closed interval [a,b], namely a and b.

Absolute Extrema

The largest or smallest values of a function over a specific interval.

Evaluate the Function

To calculate the value of a function at a specific point.

Function Value

The output of a function f(x) for a given input x.

Maximum

The highest function value within a closed interval.

Minimum

The lowest function value within a closed interval.

Continuous on [a,b]

Indicates that the function must not be discontinuous on the closed interval for the Candidates Test to apply.

Discontinuous Function

A function that has breaks, jumps, or holes at certain points.

Slope of the Tangent

The derivative of a function at a point, indicating its rate of change.

Comparing Values

The process of determining which of the computed function values is the largest or smallest.

Limit

A value that a function approaches as the input approaches some value.

Identify Domain

To determine the set of all possible input values for which the function is defined.

f′(x) = 0

Indicates a critical point where the slope of the function is horizontal.

f′(x) undefined

Indicates a critical point where the derivative does not exist.

List of Function Values

A compilation of all the function outputs from endpoints and critical points for comparison.

Largest Value

The maximum value found from evaluating the function.

Smallest Value

The minimum value found from evaluating the function.

Analyze Behavior

To examine how a function behaves as it approaches certain values or ends of an interval.

Absolute Maximum

The highest function value found among the endpoints and critical points.

Absolute Minimum

The lowest function value found among the endpoints and critical points.

Evaluate f(a) and f(b)

To compute the function values at the endpoints a and b.

Compute the derivative

Calculate f′(x) to find the critical points of the function.

Interval

A range of x-values, which can be open or closed.

Function Exists

Indicates that the function value is defined and can be computed at the critical point.

Other Methods

Techniques used when the interval is open or infinite, such as limits.

Step-by-Step Guide

A series of instructions to perform the Candidates Test correctly.

Check Endpoints

To ensure both endpoints are included in the evaluation for extrema.

Compute f(c)

To determine the function value at critical points found within the interval.

Function Values Comparison

The act of evaluating and comparing the function outputs from critical points and endpoints.