Let $g(x)=x^{3}-12 x+7$ . $\newline$ The absolute maximum value of $g$ over the closed interval $[-4,5]$ occurs at what $x$ -value? $\newline$ Choose $1$ answer: $\newline$ (A) $-4$ $\newline$ (B) $-2$ $\newline$ (C) $2$ $\newline$ (D) $5$

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Algebra 2

Conjugate root theorems

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Q. Let

g(x)=x^{3}-12 x+7

\newline

The absolute maximum value of

g

over the closed interval

[-4,5]

occurs at what

x

-value?

\newline

Choose

1

answer:

\newline

(A)

-4

\newline

(B)

-2

\newline

(C)

2

\newline

(D)

5

Calculate derivative: To find the absolute maximum value of $g(x)$ on the interval $[-4, 5]$ , we need to evaluate the function at the critical points and the endpoints of the interval. Critical points occur where the derivative $g'(x)$ is zero or undefined. $\newline$ Calculate the derivative of $g(x)$ : $g'(x) = 3x^2 - 12$ .
Find critical points: Set the derivative equal to zero to find the critical points: $3x^2 - 12 = 0$ . Solve for $x$ : $x^2 = 4$ , so $x = \pm2$ . Check if these critical points are within the interval $[-4, 5]$ .
Evaluate at points: Both critical points, $x = -2$ and $x = 2$ , are within the interval $[-4, 5]$ . $\newline$ Now, evaluate $g(x)$ at the critical points and the endpoints of the interval: $x = -4, -2, 2,$ and $5$ .
Compare for maximum: Evaluate $g(x)$ at $x = -4$ : $g(-4) = (-4)^3 - 12(-4) + 7 = -64 + 48 + 7 = -9$ .
Compare for maximum: Evaluate $g(x)$ at $x = -4$ : $g(-4) = (-4)^3 - 12(-4) + 7 = -64 + 48 + 7 = -9$ .Evaluate $g(x)$ at $x = -2$ : $g(-2) = (-2)^3 - 12(-2) + 7 = -8 + 24 + 7 = 23$ .
Compare for maximum: Evaluate $g(x)$ at $x = -4$ : $g(-4) = (-4)^3 - 12(-4) + 7 = -64 + 48 + 7 = -9$ .Evaluate $g(x)$ at $x = -2$ : $g(-2) = (-2)^3 - 12(-2) + 7 = -8 + 24 + 7 = 23$ .Evaluate $g(x)$ at $x = 2$ : $g(2) = (2)^3 - 12(2) + 7 = 8 - 24 + 7 = -9$ .
Compare for maximum: Evaluate $g(x)$ at $x = -4$ : $g(-4) = (-4)^3 - 12(-4) + 7 = -64 + 48 + 7 = -9$ .Evaluate $g(x)$ at $x = -2$ : $g(-2) = (-2)^3 - 12(-2) + 7 = -8 + 24 + 7 = 23$ .Evaluate $g(x)$ at $x = 2$ : $g(2) = (2)^3 - 12(2) + 7 = 8 - 24 + 7 = -9$ .Evaluate $g(x)$ at $x = -4$ $0$ : $x = -4$ $1$ .
Compare for maximum: Evaluate $g(x)$ at $x = -4$ : $g(-4) = (-4)^3 - 12(-4) + 7 = -64 + 48 + 7 = -9$ .Evaluate $g(x)$ at $x = -2$ : $g(-2) = (-2)^3 - 12(-2) + 7 = -8 + 24 + 7 = 23$ .Evaluate $g(x)$ at $x = 2$ : $g(2) = (2)^3 - 12(2) + 7 = 8 - 24 + 7 = -9$ .Evaluate $g(x)$ at $x = -4$ $0$ : $x = -4$ $1$ .Compare the values of $g(x)$ at $x = -4$ , $x = -4$ $4$ , $x = -4$ $5$ , and $x = -4$ $6$ to find the maximum value. $x = -4$ $7$ , $x = -4$ $8$ , $x = -4$ $9$ , $g(-4) = (-4)^3 - 12(-4) + 7 = -64 + 48 + 7 = -9$ $0$ .The maximum value is $g(-4) = (-4)^3 - 12(-4) + 7 = -64 + 48 + 7 = -9$ $1$ , which occurs at $x = -4$ $0$ .