on which interval does the function g(x)=4x - 3x^2 have an average rate of change equal to 16? x = - 4 to x…

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = g(x)$ over the interval $[a,b]$ is $\frac{g(b)-g(a)}{b - a}$.

Step2: Calculate $g(x)$ for each option

Option 1: $a=-4,b = 2$

$g(-4)=4(-4)-3(-4)^2=-16 - 48=-64$, $g(2)=4(2)-3(2)^2=8 - 12=-4$. Then $\frac{g(2)-g(-4)}{2-(-4)}=\frac{-4+64}{6}=\frac{60}{6} = 10$.

Option 2: $a=-4,b = 0$

$g(-4)=-64$, $g(0)=4(0)-3(0)^2 = 0$. Then $\frac{g(0)-g(-4)}{0-(-4)}=\frac{0 + 64}{4}=16$.

Option 3: $a=-4,b = 3$

$g(-4)=-64$, $g(3)=4(3)-3(3)^2=12-27=-15$. Then $\frac{g(3)-g(-4)}{3-(-4)}=\frac{-15 + 64}{7}=\frac{49}{7}=7$.

Option 4: $a=-4,b = 1$

$g(-4)=-64$, $g(1)=4(1)-3(1)^2=4 - 3=1$. Then $\frac{g(1)-g(-4)}{1-(-4)}=\frac{1+64}{5}=\frac{65}{5}=13$.

Answer:

None of the above (as the problem doesn't state what the average - rate - of - change value we are aiming for is. If we assume we are just calculating for each option, we have shown the calculations for each interval). If there was a target value given, we would match the result of the calculation to that value).