AP Calculus BC 5.5: Absolute Extrema

Practice AP Calculus BC questions on finding absolute extrema on a closed interval by checking critical points and endpoints.

[Maximum number: 3]

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 the value of x at which g attains an absolute minimum on the closed interval

$-6 \leq x \leq 12$ . Justify your answer.

D Find the value of x at which g attains an absolute minimum on the closed interval $-6 \leq x \leq 12$ . Justify

your answer.

For $-6 \leq x \leq 12, g$ attains a minimum either when $g^{\prime}(x)=f(x)=0$ or at an endpoint. $g^{\prime}(x)=f(x)=0$	Considers $g^{\prime}(x)=0 \quad$ Point 7 (P7)
$\begin{gathered} \Rightarrow x=0, x x -6 0 6 12 \end{gathered} \quad \begin{gathered} g(x) 0 -\frac{9 \pi}{2} \end{gathered}$	Justification Point 8 (P8)
Therefore, on the closed interval $-6 \leq x \leq 12, g$ attains an absolute minimum value at x=0.	Answer Point 9 (P9)

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FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.5 question 1