select the correct location on the image. on the interval 3, 7, at which x - value is the average rate of…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,x]$ is $\frac{f(x)-f(a)}{x - a}$. Here $a = 3$ and $f(a)=12$. Let the other $x$ - value be $x$. The average rate of change is given as 56, so we have the equation $\frac{f(x)-12}{x - 3}=56$.

Step2: Check for each $x$ - value

For $x = 4$: $\frac{f(4)-12}{4 - 3}=\frac{24 - 12}{1}=12\neq56$. For $x = 5$: $\frac{f(5)-12}{5 - 3}=\frac{28 - 12}{2}=\frac{16}{2}=8\neq56$. For $x = 6$: $\frac{f(6)-12}{6 - 3}=\frac{96 - 12}{3}=\frac{84}{3}=28\neq56$. For $x = 7$: $\frac{f(7)-12}{7 - 3}=\frac{192 - 12}{4}=\frac{180}{4}=45\neq56$. There seems to be an error in the problem setup as no value from the given table satisfies the average - rate - of - change of 56 over the interval starting from $x = 3$. But if we assume the interval is $[a,b]$ and calculate the average rate of change between two points: Let's calculate the average rate of change between consecutive points. Average rate of change between $(3,12)$ and $(4,24)$: $\frac{24 - 12}{4 - 3}=12$. Average rate of change between $(4,24)$ and $(5,28)$: $\frac{28 - 24}{5 - 4}=4$. Average rate of change between $(5,28)$ and $(6,96)$: $\frac{96 - 28}{6 - 5}=68$. Average rate of change between $(6,96)$ and $(7,192)$: $\frac{192 - 96}{7 - 6}=96$. If we consider the interval $[3,6]$: $\frac{f(6)-f(3)}{6 - 3}=\frac{96 - 12}{3}=\frac{84}{3}=28$. If we consider the interval $[3,7]$: $\frac{f(7)-f(3)}{7 - 3}=\frac{192 - 12}{4}=\frac{180}{4}=45$. If we assume the problem means to find an interval such that the average rate of change is 56. Let the two - point formula for average rate of change $\frac{f(x_2)-f(x_1)}{x_2 - x_1}=56$. Let $x_1 = 3,f(x_1)=12$. Then $f(x_2)-12=56(x_2 - 3)$. $f(x_2)=56x_2-168 + 12=56x_2-156$. From the table, we can try to find a match. If we assume we made a wrong start - point assumption and calculate the average rate of change over different intervals: Let's calculate the average rate of change over the interval $[4,7]$: $\frac{f(7)-f(4)}{7 - 4}=\frac{192 - 24}{3}=\frac{168}{3}=56$.

Answer:

The $x$ - values for the interval are $4$ and $7$. The relevant $x$ - value from the options (if we consider the non - start value of the interval) is $7$.