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1. • Rate of change of a function’s derivative

     
     • Concavity describes the rate of change of a function’s derivative. If f’ is increasing then the graph is concave up, and if f’ is decreasing, then the graph is concave down.
     calcworkshop.com/application-derivatives/second-derivative-test/

     The Second Derivative Test (How-To w/ 15 Step-by-Step Examples!)

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   How do you determine the concavity of a function?

   • The second derivative of a function f can be used to determine the concavity of the graph of f. [ 2 ] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.

   Second derivative - Wikipedia

   en.wikipedia.org/wiki/Second_derivative

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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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   What is the relationship between inflection points and concavity?

   • Describe the relationship between inflection points and concavity and how to find the inflection points of a function. Graphically, a function is concave up if its graph is curved with the opening upward (Figure 1a). Similarly, a function is concave down if its graph opens downward (Figure 1b).

   Section 2.6: Second Derivative and Concavity – Applied Calculus

   louis.pressbooks.pub/appliedcalculus/chapter/2-6-second-derivative-and-concavity/

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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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   When should we sketch a concavity?

   • Now we should sketch the concavity: when the second derivative is positive, when the second derivative is negative. Finally, we can sketch our curve: 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.

   Concavity and the 2nd Derivative Test - Ximera

   ximera.osu.edu/math160fa17/m160exam3content/maximumsAndMinimums/digInConcavityAnd2ndDerivTest

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   What does the second derivative tell us about the geometry of functions?

   • Here we examine what the second derivative tells us about the geometry of functions. We know that the sign of the derivative tells us whether a function is increasing or decreasing at some point. Likewise, the sign of the second derivative tells us whether is increasing or decreasing at .

   Concavity and the 2nd Derivative Test - Ximera

   ximera.osu.edu/math160fa17/m160exam3content/maximumsAndMinimums/digInConcavityAnd2ndDerivTest

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3. www.khanacademy.org › math › ap-calculus-abConcavity review (article) | Khan Academy

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

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

   • You have two functions f and g, where f''>0 and g''>0. If you take the second derivative of f+g, you get f''+g'', which is positive. So their sum i...
   • In order for 𝑓(𝑥) to be concave up, in some interval, 𝑓 ''(𝑥) has to be greater than or equal to 0 (i.e. non-negative) for all 𝑥 in that inter...
   • In general, no. Points where f"(x)= 0 are specifically called inflection points. But in the example provided, -2 is also a critical point (Observe...
   • Product rule: d/dx[f(x)*g(x)] = f'(x)g(x) + f(x)g'(x) f(x) = (x-1)³ and g(x) = x

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 · Interval 2, \((-1,0)\): For any number \(c\) in this interval, the term \(2c\) in the numerator will be negative, the term \((c^2+3)\) in the numerator will be positive, and the term \((c^2-1)^3\) in the denominator will be negative. Thus \(f''(c)>0\) and \(f\) is concave up on this interval.

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

   math.libretexts.org › Courses › Chabot_College

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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.

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

   louis.pressbooks.pub › appliedcalculus › chapter

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   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.

8. Videos

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     Concavity, Inflection Points, and Second Derivative

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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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     Find the intervals of concavity from the derivative graph

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9. en.wikipedia.org › wiki › Second_derivativeSecond derivative - Wikipedia

   en.wikipedia.org › wiki › Second_derivative

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   The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.

10. ximera.osu.edu › digInConcavityAnd2ndDerivTestConcavity and the 2nd Derivative Test - Ximera

    ximera.osu.edu › digInConcavityAnd2ndDerivTest

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    Concavity. We know that the sign of the derivative tells us whether a function is increasing or decreasing at some point. Likewise, the sign of the second derivative f′′(x) tells us whether f′(x) is increasing or decreasing at x. We summarize the consequences of this seemingly simple idea in the table below:

11. spot.pcc.edu › math › APEXCalculusConcavity and the Second Derivative - PCC

    spot.pcc.edu › math › APEXCalculus

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    We have identified the concepts of concavity and points of inflection. It is now time to practice using these concepts; given a function, we should be able to find its points of inflection and identify intervals on which it is concave up or down.