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1. • F ′ is increasing

     

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     • Let f be differentiable on an interval I. The graph of f is concave up on I if f ′ is increasing. The graph of f is concave down on I if f ′ is decreasing. If f ′ is constant then the graph of f is said to have no concavity.
     Reference:

     3.4: Concavity and the Second Derivative - Mathematics LibreTexts

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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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   Is the graph of (F) concave down on (I)?

   • The graph of \ (f\) is concave down on \ (I\) if \ (f'\) is decreasing. If \ (f'\) is constant then the graph of \ (f\) is said to have no concavity. Note: We often state that "\ (f\) is concave up" instead of "the graph of \ (f\) is concave up" for simplicity. The graph of a function \ (f\) is concave up when \ (f'\) is increasing.

   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 is concave?

   • When the slope continually decreases, the function is concave downward. Taking the second derivative actually tells us if the slope continually increases or decreases. When the second derivative is positive, the function is concave upward. When the second derivative is negative, the function is concave downward.

   Concave Upward and Downward - Math is Fun

   www.mathsisfun.com/calculus/concave-up-down-convex.html

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   Is (F) concave up?

   • Note: We often state that "\ (f\) is concave up" instead of "the graph of \ (f\) is concave up" for simplicity. The graph of a function \ (f\) is concave up when \ (f'\) is increasing. That means as one looks at a concave up graph from left to right, the slopes of the tangent lines will be increasing.

   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 be concave up or down?

   • A function can be concave up and either increasing or decreasing. Similarly, a function can be concave down and either increasing or decreasing. It’s probably not the best way to define concavity by saying which way it “opens” since this is a somewhat nebulous definition.

   Section 4.6 : The Shape of a Graph, Part II - Pauls Online Math Notes

   tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx

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   Can a second derivative determine concavity without a graph?

   • Fortunately, the second derivative can be used to determine the concavity of a function without a graph or the need to check every single x-value.

   Concavity - Math.net

   www.math.net/concavity

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3. 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 · The graph of a function \(f\) is concave up when \(f'\) is increasing. That means as one looks at a concave up graph from left to right, the slopes of the tangent lines will be increasing. Consider Figure \(\PageIndex{1}\), where a concave up graph is shown along with some tangent lines.

4. www.mathsisfun.com › calculus › concave-up-downConcave Upward and Downward - Math is Fun

   www.mathsisfun.com › calculus › concave-up-down

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   Taking the second derivative actually tells us if the slope continually increases or decreases. When the second derivative is positive, the function is concave upward. When the second derivative is negative, the function is concave downward.

5. www.math.net › concavityConcavity - Math.net

   www.math.net › concavity

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   If f' (x) is decreasing over an interval, then the graph of f (x) is concave down over the interval. Given a graph of f (x) or f' (x), as well as the facts above, it is relatively simple to determine the concavity of a function.

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6. Videos

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     12:49

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     Concavity, Inflection Points, and Second Derivative

     4 Mar 2018

     775.3K views

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     3:51

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     Ex: Determine Increasing / Decreasing / Concavity by Analyzing the Graph of a Function

     16 Jul 2013

     134.7K views

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     54:18

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

     29 Sep 2016

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7. 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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8. math.libretexts.org › Courses › Penn_State_University2.6: Second Derivative and Concavity - Mathematics LibreTexts

   math.libretexts.org › Courses › Penn_State_University

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   If f ′ (x) is negative on an interval, the graph of y = f(x) is decreasing on that interval. The second derivative tells us if a function is concave up or concave down. If f ″ (x) is positive on an interval, the graph of y = f(x) is concave up on that interval.

9. www.khanacademy.org › math › ap-calculus-abKhan Academy

   www.khanacademy.org › math › ap-calculus-ab

   Review your knowledge of concavity of functions and how we use differential calculus to analyze it.

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

    math.libretexts.org › Bookshelves › Calculus

    • Cached

    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.