Name: classifying critical points with the second derivative
Uploaded: 2026-01-12T12:24:53.000Z
Duration: 1 min 24 s
Channel: JoVE

Kesit alanı yüksekliğe göre değişen bir kupa düşünün—alt ve üstte daha geniş, ortası ise daha dardır.Bu fincana kahve sabit hacimsel bir hızda döküldüğünde, zamanla kahve seviyesi artar. Bu yükselişin hızı, o yükseklikteki kesit alanıyla ters ilişkilidir.Eğrinin içbükeyliği, yüksekliğin zamana göre ikinci türevinin işaretine bağlıdır.Kupanın alt yarısında, kesit alanı yüksekliğin hızlanmasına neden olacak şekilde değişir. Sıvının yüksekliği hızlandığı için, ikinci türev bu bölgede pozitiftir ve bu da içbükey bir eğri oluşturur.Öte yandan, kesitsel alan üst yarıda artar ve tam tersi bir etki gösterir: yükseklik yavaşlar, yani ikinci türev negatiftir ve grafikte içbükey bir aşağı bölgeye karşılık gelir.Çekim noktaları konkaveyin değiştiği noktayı işaretler.Bu örnekte, dönüş noktası kupayın ortasına yakın, kesit alanının minimum olduğu yerde yer alır. Yani, ikinci türeviyle temsil edilen yükseklik ivmelenmesi, pozitiften negatif değerlere geçişten sonra sıfıra düşmüştür.

İkinci türev testi: problem Çözümü

Second Derivative Test: Problem Solving

Matematiksel analizde, bir fonksiyonun davran&#x131;&#x15F;&#x131;n&#x131; anlamak i&#xE7;in en y&#xFC;ksek ve en d&#xFC;&#x15F;&#xFC;k de&#x11F;erlerinin belirlenmesi b&#xFC;y&#xFC;k &#xF6;nem ta&#x15F;&#x131;r. Kritik noktalar olarak adland&#x131;r&#x131;lan bu noktalar, birinci t&#xFC;revin s&#x131;f&#x131;r oldu&#x11F;u veya tan&#x131;ms&#x131;z oldu&#x11F;u yerlerde ortaya &#xE7;&#x131;kar. Kritik noktalar, &#x130;kinci T&#xFC;rev Testi kullan&#x131;larak s&#x131;n&#x131;fland&#x131;r&#x131;labilen yerel maksimum ve minimum adaylar&#x131;d&#x131;r. Ancak her kritik nokta mutlaka bir yerel maksimum ya da minimuma kar&#x15F;&#x131;l&#x131;k gelmez. Bu noktalar&#x131;n s&#x131;n&#x131;fland&#x131;r&#x131;labilmesi i&#xE7;in ikinci t&#xFC;rev incelenir. &#x130;kinci T&#xFC;rev Testi, fonksiyonun konkavl&#x131;&#x11F;&#x131;na ili&#x15F;kin bilgi verir:<ul><li>E&#x11F;er f&#x27;&#x27;(x) &gt; 0 ise, fonksiyon i&#xE7;e do&#x11F;ru yukar&#x131; konkavd&#x131;r; bu durumda kritik nokta yerel bir minimumdur.</li><li>E&#x11F;er f&#x27;&#x27;(x) &lt; 0 ise, fonksiyon i&#xE7;e do&#x11F;ru a&#x15F;a&#x11F;&#x131; konkavd&#x131;r; bu durumda kritik nokta yerel bir maksimumdur.</li></ul>E&#x11F;er f&#x27;&#x27;(x) = 0 ise, test sonu&#xE7; vermez ve Birinci T&#xFC;rev Testi gibi ek y&#xF6;ntemlerin uygulanmas&#x131; gerekir. A&#x15F;a&#x11F;&#x131;daki fonksiyonu inceleyelim:\begin{equation*} f(x) = x^3 -3x^2 + 4 \end{equation*}<ul><li>Birinci t&#xFC;revi bulunuz:</li></ul> \begin{equation*} f&#x27;(x) = 3x^2 -6x \end{equation*}Kritik noktalar&#x131; bulmak i&#xE7;in f&#x27;(x) = 0 e&#x15F;itli&#x11F;ini kurun. Bu ifade, x = 0 ve x = 2 noktalar&#x131;n&#x131; kritik noktalar olarak verir.<ul><li>&#x130;kinci t&#xFC;revi bulunuz:</li></ul>\begin{equation*} f&#x27;&#x27;(x) = 6x -6 \end{equation*}<ul><li>&#x130;kinci t&#xFC;revin kritik noktalardaki de&#x11F;erlerini de&#x11F;erlendiriniz:</li><li>f&#x27;&#x27;(0) = 6(0) &#x2212; 6 = &#x2212;6 (negatif oldu&#x11F;u i&#xE7;in x = 0 yerel maksimumdur).</li><li>f&#x27;&#x27;(2) = 6(2) &#x2212; 6 = 6 (pozitif oldu&#x11F;u i&#xE7;in x = 2 yerel minimumdur).</li></ul>Bir fonksiyonun ikinci t&#xFC;revinin i&#x15F;aret de&#x11F;i&#x15F;tirdi&#x11F;i noktada bir b&#xFC;k&#xFC;m noktas&#x131; (e&#x11F;rilik de&#x11F;i&#x15F;im noktas&#x131;) bulunur. f&#x27;&#x27;(x) = 0 e&#x15F;itli&#x11F;i kurulup x i&#xE7;in &#xE7;&#xF6;z&#xFC;ld&#xFC;&#x11F;&#xFC;nde x = 1 elde edilir. f&#x27;&#x27;(x) x = 1 noktas&#x131;nda i&#x15F;aret de&#x11F;i&#x15F;tirdi&#x11F;inden, bu nokta bir e&#x11F;rilik de&#x11F;i&#x15F;im noktas&#x131;d&#x131;r. Bu analiz, &#x130;kinci T&#xFC;rev Testi&#x2019;nin bir fonksiyonun grafi&#x11F;inin temel &#xF6;zelliklerinin belirlenmesine nas&#x131;l yard&#x131;mc&#x131; oldu&#x11F;unu g&#xF6;stermektedir.

İkinci Türev Testi: Problem Çözümü

Matematiksel analizde, bir fonksiyonun davran&#x131;&#x15F;&#x131;n&#x131; anlamak i&#xE7;in en y&#xFC;ksek ve en d&#xFC;&#x15F;&#xFC;k de&#x11F;erlerinin ...

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Applications of Differentiation

Türevin Uygulamaları

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

classifying critical points with the second derivative

The second derivative test helps classify critical points of a function. Critical points are places where the first derivative is zero or undefined. They are possible locations for local maxima and local minima, but they do not always become one or the other.

The second derivative shows concavity, or whether a graph bends up or down. If f''(x) &gt; 0, the function is concave up, and the critical point is a local minimum. If f''(x) &lt; 0, the function is concave down, and the critical point is a local maximum.

If f''(x) = 0, the test does not give a clear answer. In that case, another method, such as the First Derivative Test, must be used. The second derivative can also help identify an inflection point, which is where the concavity changes sign.

The example in the transcript begins by finding the first derivative and setting f'(x) = 0. This gives the critical points x = 0 and x = 2. Then the second derivative is checked at each point. Since f''(0) = -6, x = 0 is a local maximum. Since f''(2) = 6, x = 2 is a local minimum.

The same example also shows an inflection point. Setting f''(x) = 0 gives x = 1. Because the second derivative changes sign at x = 1, that point is an inflection point. This process shows how the second derivative test reveals key features of a graph.

Classifying Critical Points with the Second Derivative

classifying-critical-points-with-the-second-derivative

absolute and local extreme values

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Absolute and local extreme values show the highest and lowest points on a function graph. Absolute maximum and absolute minimum values describe the highest and lowest values across the entire domain. Local maximum and local minimum values describe the highest and lowest values within a smaller part of the graph. Periodic functions such as sine and cosine can have extreme values at many points. Their repeating pattern creates multiple local extrema. In these cases, the derivative becomes zero...

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first-derivatives-and-the-shape-of-a-graph

Using First Derivatives to Find Graph Changes

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Finding Turning Points with the First Derivative

The first derivative test is used to find turning points on a smooth curve. It shows where a function changes from increasing to decreasing, or from decreasing to increasing. In this example, the curve can model an asset price that falls to a low point, rebounds, and then gradually declines again. The process starts by finding the first derivative of the function. Because the function is a product of a polynomial term and an exponential term, the product rule is needed. After differentiating,...

second derivatives and the shape of a graph

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Using Second Derivatives to Read Curvature

The second derivative shows how a graph bends. It gives key information about curvature and how that curvature changes over an interval. These ideas help with graph analysis in math and in real-world settings such as road elevation, population growth, and economic trends. A function f(x) is concave upward on an interval when its graph lies above all of its tangent lines. In this case, the second derivative is positive. That means the slope of the tangent line is increasing, and the graph bends...

second derivative test: problem solving

The second derivative test helps classify critical points of a function. Critical points are places where the first derivative is zero or undefined. They are possible locations for local maxima and local minima, but they do not always become one or the other. The second derivative shows concavity, or whether a graph bends up or down. If f''(x) &gt; 0, the function is concave up, and the critical point is a local minimum. If f''(x) &lt; 0, the function is concave down, and the critical point is...

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slant asymptotes

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optimization problems

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Antiderivative graphs show how a function accumulates over time. In calculus, an antiderivative tracks that buildup, much like the changing position of a rolling ball as its velocity changes. The graph of the antiderivative reflects the total effect of the original function's values. The sign of the original function helps determine the antiderivative's direction. When the function is positive, the antiderivative rises because the accumulation grows. When the function is negative, the...

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Finding Car Stopping Distance with Integration

Integration connects acceleration, velocity, and displacement in linear motion. It works as the reverse of a derivative, which tells how one quantity changes with another. In physics, this makes it possible to move from a rate of change back to the motion itself. A car provides a clear example. It is moving at a steady 20 meters per second when an obstacle appears 800 meters ahead. To stop in time, the car slows with a constant acceleration. That acceleration tells how the car’s velocity...

key terms and concepts

Learn how the second derivative test classifies critical points as local maxima or minima and how it identifies inflection points from a function's graph.

Classifying Critical Points with the Second Derivative | 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

Classifying Critical Points with the Second Derivative

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