AP Calculus BC 5.6 Concavity Overview

Review concavity by analysing second-derivative signs, locating inflection points and describing changing rates on intervals shown by a graph or formula.

[Maximum number: 2]

The continuous function f is defined on the closed interval $-6 \leq x \leq 12$ . The graph of f,

consisting of two semicircles and one line segment, is shown in the figure.

$Graph of $f$$

Graph of $f$

Let g be the function defined by $g(x)=\int_{6}^{x} f(t) d t$ .

Find all values of x in the open interval -6<x<12 at which the graph of g has a point of

inflection. Give a reason for your answer.

B Find all values of x in the open interval -6<x<12 at which the graph of g has a point of inflection.

Give a reason for your answer.

The graph of g has a point of inflection where $g^{\prime \prime}=f^{\prime}$ changes sign, which is where $g^{\prime}=f$ changes from decreasing to increasing or vice versa.	Answer	Point 3 (P3)
The graph of g has points of inflection at x=-3 and x=6 because f changes from decreasing to increasing there.
The graph of g also has a point of inflection at x=3 because f changes from increasing to decreasing there.

Scoring Notes for Part B

- P3 is earned only for an answer of x=-3, x=3, and x=6. If any other/additional values of x in

-6<x<12 are declared to be points of inflection, the response does not earn either P3 or P4.

Consideration of x=-6 or of x=12 does not impact scoring.

- To earn P4, a response must tie the reason to the given graph of f.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because f changes from

increasing to decreasing or decreasing to increasing there" earns both P3 and P4.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because the slope of f

changes sign there" earns both P3 and P4.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because f attains relative

extrema there" earns both P3 and P4.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because g changes

concavity there" earns P3 but not P4.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because $g^{\prime \prime}=f^{\prime}$

changes sign there" earns P3 but not P4.

○ A response that relies upon an ambiguous term such as "the function" or "the graph" does not

earn P4.

- Special case: A response with two of the three correct x-values with correct reasoning and no

other/additional values of x declared to be points of inflection earns P4 but not P3.

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Covered syllabus points

5.6 Determining Concavity of Functions over Their Domains question 1