Name: finding absolute extrema on a closed interval
Uploaded: 2026-01-12T12:24:20.000Z
Duration: 1 min 21 s
Channel: JoVE

Bir fonksiyonun en yüksek veya en düşük değerlerine ulaştığı yerin bulunması, genel davranışını tanımlamaya yardımcı olur. Bu değerleri belirlemek, bir yürüyüş yolu boyunca zirveleri ve vadileri bulmak gibidir.Bir fonksiyon, türevi sıfır veya tanımlanmamış sayılarda bu değerlere ulaşabilir. Bu sayılara, türevanın sıfır veya tanımlanmamış olduğu alandaki sayılar kritik sayılar denir.Fermat Teoremi'ne göre, bir fonksiyonun bir sayıda yerel maksimum veya minimumu varsa ve türev orada varsa, türevin sıfır olması gerekir.Bu, yerel ekstremaların her zaman kritik sayılarda göründüğü anlamına gelir, ancak her kritik sayı bir ekstremuma karşılık gelmez.Bazı fonksiyonlarda türevi belirsiz olan keskin köşeler bulunur, ancak yerel bir uç nokta yine de bulunabilir.Kapalı bir aralıkta mutlak en yüksek veya en düşük değerleri bulmak için Kapalı Aralık Yöntemi kullanılır. Bu üç adımdan oluşur: açık aralıkta kritik sayıları belirlemek, bu sayılarda ve uç noktalarda fonksiyonu değerlendirmek. Bu sonuçların en büyük ve en küçüğü mutlak maksimumu ve minimumu verir.

kritik sayılar ve kapalı aralık yöntemi

Critical Numbers and the Closed Interval Method

Bir fonksiyonun en b&#xFC;y&#xFC;k ve en k&#xFC;&#xE7;&#xFC;k de&#x11F;erlerinin anla&#x15F;&#x131;lmas&#x131;, fonksiyonun genel davran&#x131;&#x15F;&#x131;n&#x131;n analiz edilmesi a&#xE7;&#x131;s&#x131;ndan son derece &#xF6;nemlidir. Genellikle ekstremumlar olarak adland&#x131;r&#x131;lan bu de&#x11F;erler, bir fonksiyonun tan&#x131;m k&#xFC;mesi boyunca nas&#x131;l davrand&#x131;&#x11F;&#x131;na dair &#xF6;nemli bilgiler sunar. Matematiksel a&#xE7;&#x131;dan ekstremumlar, s&#x131;n&#x131;rl&#x131; bir b&#xF6;lgede tepe ve &#xE7;ukur noktalar&#x131;n&#x131; ifade eden yerel ekstremumlar ile bir aral&#x131;k boyunca en y&#xFC;ksek veya en d&#xFC;&#x15F;&#xFC;k noktalar&#x131; g&#xF6;steren mutlak ekstremumlar olarak s&#x131;n&#x131;fland&#x131;r&#x131;l&#x131;r.
Bir fonksiyonun ekstremumlar&#x131;, t&#xFC;revin s&#x131;f&#x131;r oldu&#x11F;u ya da tan&#x131;ml&#x131; olmad&#x131;&#x11F;&#x131; tan&#x131;m k&#xFC;mesindeki kritik say&#x131;larda ortaya &#xE7;&#x131;kar. Fermat Teoremi&#x2019;ne g&#xF6;re, bir fonksiyon bir noktada yerel bir ekstremuma sahipse ve o noktada t&#xFC;revlenebilir ise, t&#xFC;revinin bu noktada s&#x131;f&#x131;r olmas&#x131; gerekir. Ancak bunun tersi zorunlu olarak do&#x11F;ru de&#x11F;ildir; di&#x11F;er bir deyi&#x15F;le, her kritik say&#x131; yerel bir ekstremuma kar&#x15F;&#x131;l&#x131;k gelmez. Baz&#x131; kritik say&#x131;lar, ekstremum niteli&#x11F;i ta&#x15F;&#x131;mayan b&#xFC;k&#xFC;m noktalar&#x131;n&#x131; veya d&#xFC;z b&#xF6;lgeleri g&#xF6;sterebilir.
Ayr&#x131;ca, kritik say&#x131;lar t&#xFC;revin tan&#x131;ml&#x131; olmad&#x131;&#x11F;&#x131; noktalar&#x131; da kapsar. Bu durumlar genellikle keskin k&#xF6;&#x15F;elere veya kusp, yani sivri u&#xE7; noktalar&#x131;na sahip fonksiyonlarda ortaya &#xE7;&#x131;kar. &#xD6;rne&#x11F;in, mutlak de&#x11F;er fonksiyonu gibi durumlarda klasik anlamda t&#xFC;revlenebilirlik sa&#x11F;lanamaz; buna ra&#x11F;men bu t&#xFC;r noktalarda yerel bir maksimum ya da minimum de&#x11F;er bulunabilir.
Bir fonksiyonun kapal&#x131; [a, b] aral&#x131;&#x11F;&#x131;nda mutlak ekstremumlar&#x131;n&#x131; belirlemek i&#xE7;in Kapal&#x131; Aral&#x131;k Y&#xF6;ntemi kullan&#x131;l&#x131;r. Bu y&#xF6;ntem &#xFC;&#xE7; ad&#x131;mdan olu&#x15F;ur:
<ul>
<li>(a, b) a&#xE7;&#x131;k aral&#x131;&#x11F;&#x131;ndaki t&#xFC;m kritik say&#x131;lar&#x131;n belirlenmesi</li>
<li>Fonksiyonun bu kritik say&#x131;lardaki de&#x11F;erlerinin hesaplanmas&#x131;</li>
<li>Fonksiyonun u&#xE7; noktalar a ve b&#x2019;deki de&#x11F;erlerinin hesaplanmas&#x131;</li>
</ul>
Elde edilen bu de&#x11F;erler kar&#x15F;&#x131;la&#x15F;t&#x131;r&#x131;ld&#x131;&#x11F;&#x131;nda, en b&#xFC;y&#xFC;k de&#x11F;er bu aral&#x131;kta mutlak maksimum, en k&#xFC;&#xE7;&#xFC;k de&#x11F;er ise bu aral&#x131;kta mutlak minimum olarak belirlenir. Bu sistematik yakla&#x15F;&#x131;m, t&#xFC;m olas&#x131; ekstremum noktalar&#x131;n&#x131;n dikkate al&#x131;nmas&#x131;n&#x131; sa&#x11F;layarak, bir fonksiyonun belirli bir tan&#x131;m aral&#x131;&#x11F;&#x131;ndaki ekstremal davran&#x131;&#x15F;&#x131;n&#x131;n kapsaml&#x131; bir bi&#xE7;imde incelenmesine olanak tan&#x131;r.

Kritik Sayılar ve Kapalı Aralık Yöntemi

Bir fonksiyonun en b&#xFC;y&#xFC;k ve en k&#xFC;&#xE7;&#xFC;k de&#x11F;erlerinin anla&#x15F;&#x131;lmas&#x131;, fonksiyonun genel ...

critical-numbers-and-the-closed-interval-method

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Applications of Differentiation

Türevin Uygulamaları

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

finding absolute extrema on a closed interval

Absolute extrema on a closed interval are found by checking critical numbers and the endpoints. A function’s maximum and minimum values help describe its overall behavior across its domain. These values are called extrema, and they can be local or absolute.

Local extrema are peaks and valleys in a limited region. Absolute extrema are the highest or lowest points on the entire interval. Critical numbers are the domain values where the derivative is zero or does not exist. According to Fermat’s Theorem, if a function has a local extremum at a point where it is differentiable, then the derivative at that point must be zero.

A critical number does not always mean there is a local extremum. Some critical numbers mark inflection points or flat regions instead. Critical numbers also include points where the derivative is undefined. This can happen at sharp corners or cusps, such as on an absolute value function, where a local maximum or minimum may still occur.

The Closed Interval Method is used to find absolute extrema on an interval written as [a, b]. First, identify all critical numbers in the open interval (a, b). Next, evaluate the function at those critical numbers. Then evaluate the function at the endpoints a and b.

After all of these values are found, compare them. The greatest value is the absolute maximum, and the least value is the absolute minimum on the interval. This method makes sure every possible extremal point is checked, giving a complete picture of the function’s behavior on a closed interval.

Finding Absolute Extrema on a Closed Interval

finding-absolute-extrema-on-a-closed-interval

absolute and local extreme values

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critical numbers and the closed interval method

Absolute extrema on a closed interval are found by checking critical numbers and the endpoints. A function’s maximum and minimum values help describe its overall behavior across its domain. These values are called extrema, and they can be local or absolute. Local extrema are peaks and valleys in a limited region. Absolute extrema are the highest or lowest points on the entire interval. Critical numbers are the domain values where the derivative is zero or does not exist. According to Fermat’s...

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application-of-differentiation-to-business

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key terms and concepts

Learn how to find absolute maximum and minimum values on a closed interval by checking critical numbers, endpoints, and the Closed Interval Method.

Finding Absolute Extrema on a Closed Interval | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Absolute Extrema on a Closed Interval

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