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, $a=-1$, $b = 3$, and $f(x)=x^{4}-2x$.

Step2: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)$: $f(-1)=(-1)^{4}-2\times(-1)=1 + 2=3$.

Step3: Calculate $f(3)$

Substitute $x = 3$ into $f(x)$: $f(3)=3^{4}-2\times3=81-6 = 75$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(3)-f(-1)}{3-(-1)}$, we substitute $f(-1)=3$ and $f(3)=75$: $\frac{75 - 3}{3+1}=\frac{72}{4}=18$.

Answer:

18