How To Find Absolute Minimum Value Of A Function

Shaun Ault

AP Calculus Review: Finding Accented Extrema

By on March 3, 2017 in AP

The absolute extrema of a part are the largest and smallest values of the function. What is the most turn a profit that a company can make? What is the least amount of fence needed to enclose a garden? Once you know how to find the absolute extrema of a role, and so y'all can answer these kinds of questions and many more than!

Overview: What are Accented Extrema?

The absolute extrema of a role f on a given domain set up D are the accented maximum and absolute minimum values of f(x) as x ranges throughout D.

In other words, we say that Yard is the accented maximum if M = f(c) for some c in D, and f(ten) ≤ M for all other x in D.

Nosotros define the absolute minimum chiliad in much the same way, except that f(ten) ≥ m for all x in D.

Graph showing absolute maximum and minimum points

Functions with Discontinuity

Sometimes a part may fail to have an absolute minimum or maximum on a given domain set. This often happens when the office has a aperture.

Graph with a vertical asymptote. It has an absolute minimum but no absolute maximum.

This function is discontinuous on the interval shown. It has an absolute minimum value, 0, but no absolute maximum.

Domain Sets and Extrema

Even if the office is continuous on the domain prepare D, there may be no extrema if D is not closed or divisional.

For example, the parabola function, f(x) = x ² has no absolute maximum on the domain set (-∞, ∞). This is because the values of x ² keep getting larger and larger without bound every bit x → ∞. By the way, this role does accept an absolute minimum value on the interval: 0.

Nonetheless, there may yet be issues fifty-fifty on a divisional domain gear up. The function beneath has neither absolute minimum nor maximum because the endpoints of the interval are not in its domain. Note, the open up circles on the graph mean that those points are missing, so in that location cannot be any extrema at those points.

Graph defined on an open interval. No absolute extrema.

This graph is defined on the open interval, (-4, 4). There are no absolute extrema.

The Farthermost Value Theorem

In do, we normally require D to be a closed interval of the form [a, b] for some constants a < b. In that case, the Farthermost Value Theorem guarantees both absolute extrema must exist.

The Extreme Value Theorem (EVT) If a office f is continuous on a closed, bounded interval [a, b], and then f attains both absolute extrema on that interval.

Only be careful: the EVT simply works in one direction. If the function is continuous on a closed, divisional interval, then it must take absolute extrema on that interval. However a function may fail to encounter the weather of the EVT and still have an accented maximum and/or minimum.

Finding the Absolute Extrema

In the case that f is continuous on [a, b], then the post-obit procedure will locate the absolute extrema.

The Closed Interval Method

The method requires computing a derivative. If you need a refresher, check out this Calculus Review: Derivative Rules.

Find all critical numbers of f within the interval [a, b]. That is, set f '(x) = 0, solve for x, and only consider those solutions x that satisfy a ≤ 10 ≤ b.
Plug in each disquisitional number from pace 1 into the role f(10).
Plug in the endpoints, a and b, into the function f(x).
The largest value is the accented maximum, and the smallest value is the absolute minimum.

Instance

Let'due south find the absolute extrema of f(x) = 10 ³ – 12x + 23 on the interval [-v, 3].

Because f is continuous on [-5, iii], which is a closed and divisional interval, the EVT guarantees both an absolute maximum and minimum must be on the given interval. Furthermore, we tin can using the Airtight Interval Method to notice them.

Step 1: Commencement find the disquisitional numbers.

f '(x) = 3x ² – 12 = 3(x ² – 4) = 3(ten – two)(10 + 2)

Setting 3(x – two)(x + ii) = 0, nosotros find two critical numbers: -2 and two, both of which are in the given interval.

Steps 2 and 3: I tend to combine these steps in my work. Build a tabular array of x-values on the left, including the critical numbers and the endpoints of the interval. And then in the correct column, plug each one into f(x).

x	f(x) = x ³ - 12x + 23
-5	-42	Min
-ii	39	Max
ii	7
3	14

The absolute maximum value is 39 (at x = -two), and the absolute minimum is -42 (at x = -v).

Looking at the graph of f, you can verify the max and min values.

Absolute extrema illustrated on a graph (example)

Summary

The absolute extrema of a role on a given domain prepare D are the greatest and to the lowest degree values of the function on D.
The Extreme Value Theorem guarantees that a continuous function must have absolute extrema on a bounded, airtight interval.
You can employ the Closed Interval Method to locate the absolute extrema.

Now that you know more about accented extrema, yous can maximize your score on the AP Calculus exams!

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer scientific discipline from Oberlin College in 2002. In add-on, Shaun earned a B. Mus. from the Oberlin Solarium in the same year, with a major in music composition. Shaun still loves music -- almost as much equally math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!

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