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$The derivative of a function h is given by h^(')(x)=x^(2)-10 sin(x). On which intervals is the graph of h increasing? Use a graphing calculator. Choose 1 answer: (B) {:[(-oo","-3.837]" and "],[[-1.977","1.306]]:} (C) (-oo,0] and [2.479,oo) (D) [0,2.479] (E) All real numbers$

The derivative of a function $h$ is given by $h^{\prime}(x)=x^{2}-10 \sin (x)$ . $\newline$ On which intervals is the graph of $h$ increasing? $\newline$ Use a graphing calculator. $\newline$ Choose $1$ answer: $\newline$ (B) $\begin{array}{l}(-\infty,-3.837] \text { and } \\ {[-1.977,1.306]}\end{array}$ $\newline$ (C) $(-\infty, 0]$ and $[2.479, \infty)$ $\newline$ (D) $[0,2.479]$ $\newline$ (E) All real numbers

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Find derivatives of sine and cosine functions

Full solution

Q. The derivative of a function

h

is given by

h^{\prime}(x)=x^{2}-10 \sin (x)

.

\newline

On which intervals is the graph of

h

increasing?

\newline

Use a graphing calculator.

\newline

Choose

1

answer:

\newline

(B)

\begin{array}{l}(-\infty,-3.837] \text { and } \\ {[-1.977,1.306]}\end{array}

\newline

(C)

(-\infty, 0]

and

[2.479, \infty)

\newline

(D)

[0,2.479]

\newline

(E) All real numbers

Find where $h$ is increasing: To find where $h$ is increasing, we need to find where h'(x) > 0.
Set $h'(x)$ condition: Set $h'(x) = x^2 - 10 \sin(x)$ greater than $0$ to find the intervals. $\newline$ x^2 - 10 \sin(x) > 0
Graph $y = x^2 - 10 \sin(x)$ : Use a graphing calculator to plot $y = x^2 - 10 \sin(x)$ and find the $x$ -values where the graph is above the $x$ -axis.
Intervals above x-axis: The calculator shows the graph of $y = x^2 - 10 \sin(x)$ is above the x-axis in the intervals $(-\infty, -3.837]$ and $[-1.977, 1.306]$ .

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