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1. • • A graph is concave up where its second derivative is positive and concave down where its second derivative is negative. Thus, the concavity changes where the second derivative is zero or undefined. Such a point is called a point of inflection.
     teachingcalculus.com/2012/10/24/concavity/

     Concavity - Teaching Calculus

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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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   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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   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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   Does concavity change at a point?

   • At points a and h, the graph is concave up on both sides, so the concavity does not change. At points c and f, the graph is concave down on both sides. At point e, even though the graph looks strange there, the graph is concave down on both sides – the concavity does not change. Inflection points happen when the concavity changes.

   4.2: Second Derivative and Concavity - Mathematics LibreTexts

   math.libretexts.org/Courses/Chabot_College/MTH_15:_Applied_Calculus_I/04:_Applications_of_Differentiation/4.02:_Second_Derivative_and_Concavity

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   How do you know if a function is concave up or down?

   • Graphically, a function is concave up if its graph is curved with the opening upward (Figure 4.2.1a). Similarly, a function is concave down if its graph opens downward (Figure 4.2.1b). This figure shows the concavity of a function at several points. Notice that a function can be concave up regardless of whether it is increasing or decreasing.

   4.2: Second Derivative and Concavity - Mathematics LibreTexts

   math.libretexts.org/Courses/Chabot_College/MTH_15:_Applied_Calculus_I/04:_Applications_of_Differentiation/4.02:_Second_Derivative_and_Concavity

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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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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. math.libretexts.org › Courses › Chabot_College4.2: Second Derivative and Concavity - Mathematics LibreTexts

   math.libretexts.org › Courses › Chabot_College

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   Graphically, it is clear that the concavity of f(x) = x3 and h(x) = x1 / 3 changes at (0,0), so (0,0) is an inflection point for f and h. The function g(x) = x4 is concave up everywhere so (0,0) is not an inflection point of g. We can also compute the second derivatives and check the sign change.

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

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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.

9. 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.

10. louis.pressbooks.pub › appliedcalculus › chapterSection 2.6: Second Derivative and Concavity – Applied Calculus

    louis.pressbooks.pub › appliedcalculus › chapter

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    Learning Objectives. By the end of this section, the student should be able to: Describe how the second derivative of a function relates to its concavity and how to apply the second derivative test. Describe the relationship between inflection points and concavity and how to find the inflection points of a function. Second Derivative and Concavity.

11. mathstat.slu.edu › ~may › ExcelCalculusThe Second Derivative and Concavity - Saint Louis University

    mathstat.slu.edu › ~may › ExcelCalculus

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    The second derivative tells whether the curve is concave up or concave down at that point. If the second derivative is positive at a point, the graph is bending upwards at that point. Similarly, if the second derivative is negative, the graph is concave down.