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Consider the function $\newline$ $f(x)=x^{3}-9x^{2}-21 x-2.$ $\newline$ Its derivative is $\newline$ $f^{\prime}(x)=3x^{2}-18 x-21.$ help (formulas) $\newline$ its second derivative is $\newline$ $f^{\prime\prime}(x)=6x-18$ $\newline$ The graph that correctly displays $\newline$ $f(z)$ (in black) and $\newline$ $f^{\prime\prime}(x)$ (in orange) is $\newline$ A $\newline$ B $\newline$ c $\newline$ $0$ . $\newline$ B. $\newline$ Using the graphs of $\newline$ $f$ and $\newline$ $f^{\prime\prime}$ , indicate where $\newline$ $f$ is concave up and concave down. Enter DNE if such an interval does not exist: $\newline$ $f$ is concave up on $\newline$ ◻ help (intervals). $\newline$ $f$ is concave down on

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Compare linear and exponential growth

Full solution

Q. Consider the function

\newline

f(x)=x^{3}-9x^{2}-21 x-2.

\newline

Its derivative is

\newline

f^{\prime}(x)=3x^{2}-18 x-21.

help (formulas)

\newline

its second derivative is

\newline

f^{\prime\prime}(x)=6x-18

\newline

The graph that correctly displays

\newline

f(z)

(in black) and

\newline

f^{\prime\prime}(x)

(in orange) is

\newline

A

\newline

B

\newline

c

\newline

0

.

\newline

B.

\newline

Using the graphs of

\newline

f

and

\newline

f^{\prime\prime}

, indicate where

\newline

f

is concave up and concave down. Enter DNE if such an interval does not exist:

\newline

f

is concave up on

\newline

◻ help (intervals).

\newline

f

is concave down on

Identify Second Derivative: Identify the second derivative of the function $f(x)$ . The second derivative of the function $f(x)$ is given as $f''(x) = 6x - 18$ .
Find Critical Points: Find the critical points of the second derivative. $\newline$ To find the intervals of concavity, we need to determine where the second derivative changes sign. This occurs at the critical points of the second derivative. Set $f''(x) = 0$ and solve for $x$ : $\newline$ $6x - 18 = 0$ $\newline$ $6x = 18$ $\newline$ $x = \(3\)
Test Intervals for Concavity: Test the intervals around the critical point to determine concavity.\(\newline\)We have one critical point, \(x = 3\). We will test the intervals \((-\infty, 3)\) and \((3, \infty)\) to see where the second derivative is positive (indicating concave up) and where it is negative (indicating concave down).\(\newline\)Choose a test point \(x = 2\) for the interval \((-\infty, 3)\):\(\newline\)\(f''(2) = 6(2) - 18 = 12 - 18 = -6\)\(\newline\)Since \(f''(2) < 0\), the function is concave down on the interval \((-\infty, 3)\).\(\newline\)Choose a test point \(x = 4\) for the interval \((3, \infty)\):\(\newline\)\((-\infty, 3)\)\(0\)\(\newline\)Since \((-\infty, 3)\)\(1\), the function is concave up on the interval \((3, \infty)\).

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