determine the average rate of change of the following function between the given values of the variable…

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-3$, $b = 2$, and $f(x)=x^{4}+x$.

Step2: Calculate $f(-3)$

Substitute $x=-3$ into $f(x)$: [ \begin{align*} f(-3)&=(-3)^{4}+(-3)\ &=81 - 3\ &=78 \end{align*} ]

Step3: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: [ \begin{align*} f(2)&=2^{4}+2\ &=16 + 2\ &=18 \end{align*} ]

Step4: Calculate average rate of change

[ \begin{align*} \frac{f(2)-f(-3)}{2-(-3)}&=\frac{18 - 78}{2 + 3}\ &=\frac{-60}{5}\ &=- 12 \end{align*} ]

Answer:

$-12$