The second derivative of a function g is given by g^('')(x)=x^(4)-x^(2)-x". " On which interval is the graph of g concave down? Use a graphing calculator. Choose 1 answer: (A) (0,1.325) (B) (-oo,0) and (1.325,oo) (C) (-oo,0.885) (D) (0.885,oo) (E) All real numbers

Identify Concave Down: To find where the graph of g is concave down, we need to find where its second derivative g''(x) is less than 0.Calculate Second Derivative: g''(x) = x^4 - x^2 - x. Set g''(x) < 0 and solve for x.Use Graphing Calculator: We can't factor this easily, so we'll use a graphing calculator to find the intervals where g''(x) is negative.Analyze Negative Intervals: After graphing g''(x) = x^4 - x^2 - x, we see that the function is negative between two points, let's call them a and b.Determine Interval for Concavity: Using the graphing calculator, we find that the function changes from positive to negative at x ≈ 0.885 and from negative to positive at x ≈ 1.325.Therefore, the graph of g is concave down on the interval (0.885, 1.325).

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The second derivative of a function
g is given by

g^('')(x)=x^(4)-x^(2)-x". "
On which interval is the graph of
g concave down?
Use a graphing calculator.
Choose 1 answer:
(A)
(0,1.325)
(B)
(-oo,0) and
(1.325,oo)
(C)
(-oo,0.885)
(D)
(0.885,oo)
(E) All real numbers

The second derivative of a function $g$ is given by $\newline$ $g^{\prime \prime}(x)=x^{4}-x^{2}-x \text {. }$ $\newline$ On which interval is the graph of $g$ concave down? $\newline$ Use a graphing calculator. $\newline$ Choose $1$ answer: $\newline$ (A) $(0,1.325)$ $\newline$ (B) $(-\infty, 0)$ and $(1.325, \infty)$ $\newline$ (C) $(-\infty, 0.885)$ $\newline$ (D) $(0.885, \infty)$ $\newline$ (E) All real numbers

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Full solution

Q. The second derivative of a function

g

is given by

\newline

g^{\prime \prime}(x)=x^{4}-x^{2}-x \text {. }

\newline

On which interval is the graph of

g

concave down?

\newline

Use a graphing calculator.

\newline

Choose

1

answer:

\newline

(A)

(0,1.325)

\newline

(B)

(-\infty, 0)

and

(1.325, \infty)

\newline

(C)

(-\infty, 0.885)

\newline

(D)

(0.885, \infty)

\newline

(E) All real numbers

Identify Concave Down: To find where the graph of $g$ is concave down, we need to find where its second derivative $g''(x)$ is less than $0$ .
Calculate Second Derivative: $g''(x) = x^4 - x^2 - x$ . Set g''(x) < 0 and solve for $x$ .
Use Graphing Calculator: We can't factor this easily, so we'll use a graphing calculator to find the intervals where $g''(x)$ is negative.
Analyze Negative Intervals: After graphing $g''(x) = x^4 - x^2 - x$ , we see that the function is negative between two points, let's call them $a$ and $b$ .
Determine Interval for Concavity: Using the graphing calculator, we find that the function changes from positive to negative at $x \approx 0.885$ and from negative to positive at $x \approx 1.325$ .
Determine Interval for Concavity: Using the graphing calculator, we find that the function changes from positive to negative at $x \approx 0.885$ and from negative to positive at $x \approx 1.325$ .Therefore, the graph of $g$ is concave down on the interval $(0.885, 1.325)$ .