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AP Calculus AB AP Exam Review Free Response 3
This Question is *
^(**) CALCULATOR INACTIVE**
Please show all work on page
2&3
The function
g is defined and differentiable on the closed interval
[-6,6] and satisfies
g(0)=4. The graph of
y=g^(')(x), the derivative of
g, consists of a semicircle and three line segments, as shown in the figure below.
(b) Find the
x-coordinate of each point of inflection of the graph
y=g(x) on the interval
-6 < x < 6. Explain your reasoning.

AP Calculus AB AP Exam Review Free Response $3$ $\newline$ This Question is * ${ }^{*}$ CALCULATOR INACTIVE** $\newline$ Please show all work on page $2 \& 3$ $\newline$ The function $g$ is defined and differentiable on the closed interval $[-6,6]$ and satisfies $g(0)=4$ . The graph of $y=g^{\prime}(x)$ , the derivative of $g$ , consists of a semicircle and three line segments, as shown in the figure below. $\newline$ (b) Find the $x$ -coordinate of each point of inflection of the graph $y=g(x)$ on the interval $-6<x<6$ . Explain your reasoning.

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Transformations of absolute value functions: translations and reflections

Full solution

Q. AP Calculus AB AP Exam Review Free Response

3

\newline

This Question is *

{ }^{*}

CALCULATOR INACTIVE**

\newline

Please show all work on page

2 \& 3

\newline

The function

g

is defined and differentiable on the closed interval

[-6,6]

and satisfies

g(0)=4

. The graph of

y=g^{\prime}(x)

, the derivative of

g

, consists of a semicircle and three line segments, as shown in the figure below.

\newline

(b) Find the

x

-coordinate of each point of inflection of the graph

y=g(x)

on the interval

-6<x<6

. Explain your reasoning.

Identify Sign Changes: Identify where the second derivative, $g''(x)$ , changes sign, which indicates a potential point of inflection. The graph of $y=g'(x)$ is given, so we need to analyze where $g''(x) = 0$ or is undefined by looking at changes in the slope of $g'(x)$ .
Analyze Slope Graph: Examine the graph of $y=g'(x)$ . Points where the slope of $g'(x)$ changes are where $g''(x) = 0$ . Since $g'(x)$ consists of a semicircle and three line segments, focus on the endpoints of these segments and any points where the curvature changes.
Calculate Candidate Points: Calculate the $x$ -coordinates of the endpoints of the line segments and the points where the semicircle meets the line segments. These are likely candidates for points where $g''(x)$ changes sign.
Verify Sign Changes: Verify each candidate point by checking the sign of $g''(x)$ before and after each point. If the sign changes, confirm it as a point of inflection.

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