Answer
$$
y=f(x)=-2x^{3}-3x^{2}+8 \quad \text { from} \quad x=-2\quad \text { to} \quad x=6
$$
The average rate of change of $ f(x) $ with respect to $x$ as $x$ changes from $a=-2 $ to $ b=6$ is
$$
\begin{split}
\text {Average rate of change}
&=\frac{f(b)-f(a)}{b-a}\\
& =\frac{f(6)-f(-2)}{6-(-2)}\\
& =-68\\
\end{split}
$$
Instantaneous rate of change at $x= -2$:
$$
y^{\prime}(x)=-6(-2)^{2}-6(-2)=6(4)+12=-12
$$
Work Step by Step
$$
y=f(x)=-2x^{3}-3x^{2}+8 \quad \text { from} \quad x=-2\quad \text { to} \quad x=6
$$
The average rate of change of $ f(x) $ with respect to $x$ as $x$ changes from $a=-2 $ to $ b=6$ is
$$
\begin{split}
\text {Average rate of change}
&=\frac{f(b)-f(a)}{b-a}\\
& =\frac{f(6)-f(-2)}{6-(-2)}\\
& =\frac{(-2(6)^{3}-3(6)^{2}+8 )-(-2(-2)^{3}-3(-2)^{2}+8 )}{8}\\
& =\frac{(-532-12)}{8}\\
& =\frac{(-544)}{8}\\
& =-68\\
\end{split}
$$
and we can find that :
$$
y^{\prime}(x)=-6x^{2}-6x
$$
Instantaneous rate of change at $x= -2$:
$$
y^{\prime}(x)=-6(-2)^{2}-6(-2)=6(4)+12=-12
$$
