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1. • • We say that a graph is concave up if the line between two points is above the graph, or alternatively if the first derivative is increasing. (In finance, such a curve is said to be convex.) Similarly, we say that a graph is concave down if the line between two points is below the graph, or alternatively if the first derivative is decreasing.
     mathstat.slu.edu/~may/ExcelCalculus/sec-4-5-SecondDerivativeConcavity.html

     The Second Derivative and Concavity - Saint Louis University

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   Why do we need to know where a graph is concave?

   • If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.

   5.4: Concavity and Inflection Points - Mathematics LibreTexts

   math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/05:_Curve_Sketching/5.04:_Concavity_and_Inflection_Points

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   Can a function present concave and convex parts in the same graph?

   • The functions, however, can present concave and convex parts in the same graph, for example, the function f (x) = (x + 1) 3 − 3 (x + 1) 2 + 2 presents concavity in the interval (− ∞, 0) and convexity in the interval (0, ∞): The study of the concavity and convexity is done using the inflection points.

   Concavity and convexity, inflection points of a function

   www.sangakoo.com/en/unit/concavity-and-convexity-inflection-points-of-a-function

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   What is the connection between concavity and second derivative?

   • There is a connection between the concavity of the graph of a function (Definition 4.12) and its second derivative. = y x is concave down over its entire domain. point on a graph where the concavity of the curve changes (from concave down to concave up, or vice versa) is called a point of inflection (Definition 4.14).

   Calculus 140, section 4.7 Concavity and Inflection Points - UMD

   math.umd.edu/~tjp/140 04.7 lecture notes.pdf

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   Which graph represents a local minimum if a graph is concave?

   • Figure \ (\PageIndex {13}\): A graph of \ (f (x)\) in Example \ (\PageIndex {4}\). The second derivative is evaluated at each critical point. When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum.

   3.4: Concavity and the Second Derivative - Mathematics LibreTexts

   math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/03:_The_Graphical_Behavior_of_Functions/3.04:_Concavity_and_the_Second_Derivative

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   How do you know if a function f(x) is concave?

   • On the other hand, it is said that a function f (x) is concave if the function − f (x) is convex, or in other words, if the segments that join the points of the graph f (x) are all placed below the graph.

   Concavity and convexity, inflection points of a function

   www.sangakoo.com/en/unit/concavity-and-convexity-inflection-points-of-a-function

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   What is an example of a concave function?

   • Let's see an example of concave function: Commonly, we can say that the convex functions are curved functions that are first decreasing and afterwards increasing, while the concave functions are the other way round, they are first increasing and afterwards increasing.

   Concavity and convexity, inflection points of a function

   www.sangakoo.com/en/unit/concavity-and-convexity-inflection-points-of-a-function

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4. math.libretexts.org › Bookshelves › Calculus3.4: Concavity and the Second Derivative - Mathematics LibreTexts

   math.libretexts.org › Bookshelves › Calculus

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   Dec 21, 2020 · When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum. We have been learning how the first and second derivatives of a function relate information about the graph of that function.

5. math.libretexts.org › Bookshelves › Calculus5.4: Concavity and Inflection Points - Mathematics LibreTexts

   math.libretexts.org › Bookshelves › Calculus

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   Dec 21, 2020 · If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.

6. tutorial.math.lamar.edu › ShapeofGraphPtIICalculus I - The Shape of a Graph, Part II

   tutorial.math.lamar.edu › ShapeofGraphPtII

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   Nov 16, 2022 · The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points (i.e. where concavity changes) that a function may have.

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     Concavity, Inflection Points, and Second Derivative

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     Ex: Concavity / Points of Inflection by Analyzing a Graph (Algebra Topic)

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     Concavity, Inflection Points, Increasing Decreasing, First & Second Derivative - Calculus

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8. www.sangakoo.com › en › unitConcavity and convexity, inflection points of a function

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   Concavity and convexity. It is said that a function f (x) is convex if, once having joined any two points of the graph, the segment stays over the graph: In this graph we can observe different segments (with different colors) that join two points of the graph and stay over it.

9. courses.lumenlearning.com › calculus1 › chapterThe First Derivative Test and Concavity | Calculus I

   courses.lumenlearning.com › calculus1 › chapter

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   State the first derivative test for critical points. Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph. Explain the concavity test for a function over an open interval.

10. www.whitman.edu › mathematics › calculus_online5.4 Concavity and inflection points - Whitman College

    www.whitman.edu › mathematics › calculus_online

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    If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.

11. math.umd.edu › ~tjp › 140 04Calculus 140, section 4.7 Concavity and Inflection Points - UMD

    math.umd.edu › ~tjp › 140 04

    A point on a graph where the concavity of the curve changes (from concave down to concave up, or vice versa) is called a point of inflection (Definition 4.14). By implication (think about what separates positive and negative numbers on a number line), if a point (c, f (c)) is a point of inflection, then f ′ (c) = 0. Important: This is a one ...