Learning Objectives:
- What Is Absolute Extrema?
- Difference Between Local and Absolute Extrema
- Can Endpoints be Absolute Extrema?
- How to Find Absolute Extrema on a Closed Interval
- How to Find Absolute Extrema on an Open Interval
- Examples of Absolute Extrema
What Is Absolute Extrema?
Absolute extrema of a function are the absolute maximum (highest) and absolute minimum (lowest) values of a function on a given set or domain. In simpler terms, absolute extrema are a function’s highest and lowest points in the given set of domains.
A function $f$ has an absolute maximum (or global maximum) at $c$ if $f(c) \geq f(x)$ for all x in the domain.
At the same time, a function $f$ has an absolute minimum at $c$ if $f(c) \leq f(x)$ for all x in the domain.
The maximum and minimum values of $f$ are called the extreme values of $f$ or the absolute extrema.
Difference Between Local and Absolute Extrema
But what is the difference between local and absolute extrema? Local extrema (relative) is a function’s maximum and minimum value within a specific interval. In contrast, the absolute extrema (global) is a function’s maximum and minimum value over its entire domain.
Let’s graph a sample function f(x) with an absolute maximum at $d$ and an absolute minimum at $a$.
Please note that when comparing local vs absolute extrema, they have one thing in common – the function’s derivative is zero at both local and absolute extrema.
Can Endpoints be Absolute Extrema?
Given a closed interval, can endpoints be absolute extrema? Yes, there is a probability that the absolute extrema on a closed interval $[a,b]$ occurs at either endpoint given. This results in the Extreme Value Theorem stated below.
Extreme Value Theorem: Any continuous function $f$ over a closed interval $[a,b]$ has an absolute maximum and a minimum on $[a,b]$.
But remember, it is not always the case! A function can contain several inflection points, called local minima and maxima, which are good candidates to be the absolute extrema.
These points occur in the graph with a change in slope, whereas the tangent line is horizontal or vertical, suggesting a zero derivative or undefined, respectively.
It is useful to identify the critical numbers to locate the absolute extrema. Remember that critical numbers are points within the interval where absolute extrema may occur. Only a finite number of critical numbers $c_{1} c_{2}, c_{3}…..c_{k}$ exist.
What is the critical number ‘$c$’? A critical number of a function is a number $c$ in the domain of $f$ for which $f’ (c) = 0$ or $f’ (c)$ is undefined. If $f$ has a local maximum or minimum at $c$, then $c$ is a critical number of $f$.
Furthermore, let’s discuss the step-by-step method on how to find the absolute extrema of a function on a closed interval and open interval.
How to Find Absolute Extrema on Closed Interval
In a closed interval method, let $f$ be a continuous function on a closed interval $[a,b]$. To find an absolute maximum or minimum of a continuous function on a closed interval, we note that it is either a local or critical number $c$ or occurs at the endpoints $[a,b]$.
Below is the four-step procedure for finding the absolute extrema on a closed interval.
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Step 1: Obtain the function’s first derivative $f’ (c)$.
Step 2: Evaluate the function $f$ at the closed interval’s endpoints [a, b].
Step 3: Evaluate the function at critical numbers $c$ within the interval. This step is similar to finding the local extrema or relative extrema. To evaluate critical numbers, let $f’ (c)$ of the function from step 1 be equal to 0.
Note: Determine whether the critical numbers obtained lie between the closed intervals. If not within the interval, scratch it from the succeeding steps to save time.
Step 4: Compare the values obtained from steps 2 and 3. You may construct a table to easily identify the absolute maximum (the largest value) and minimum (the smallest value) values.
How to Find Absolute Extrema on Open Interval
To find an absolute maximum or minimum of a continuous function on an open interval, we note that it is either a local or critical number $c$ or occurs at the endpoints $(a,b)$ if the function is defined at the boundary points.
Recall that an open interval is a domain set of numbers within a specific interval $(a,b)$ that does not include its endpoints, which means $a < x < b$. Consider the function’s behavior near the endpoints in an open interval.
Below is the five-step procedure for finding the absolute extrema on a closed interval.
Step 1: Obtain the function’s first derivative $f’ (c)$.
Step 2: Evaluate the function $f$ at endpoints $(a, b)$ of the open interval if applicable.
Step 3: Evaluate the function at critical numbers $c$ within the interval. This step is similar to finding the local extrema or relative extrema. To evaluate critical numbers, let $f’ (c)$ of the function from step 1 be equal to 0.
Step 4: Evaluate limits at boundaries as x approaches the boundary points $(a,b)$.
Step 5: Compare the values obtained from steps 2, 3, and 4. You may construct a table to easily identify the absolute maximum (the largest value) and minimum values (the smallest value).
