Final answer:
The average rate of change of the function g(x) over the interval 2 < x < 7 is calculated by taking the difference in the function values at x=7 and x=2, and dividing it by the change in x, resulting in -12.
Step-by-step explanation:
The question asks to determine the average rate of change of the function g(x) = -2x^2 + 6x + 15 over the interval 2 < x < 7. To find the average rate of change, we calculate the difference in the function values at the endpoints of the interval and divide by the difference in the x-values.
First, calculate the function values at the endpoints:
g(2) = -2(2)^2 + 6(2) + 15 = -8 + 12 + 15 = 19
g(7) = -2(7)^2 + 6(7) + 15 = -98 + 42 + 15 = -41
Then, calculate the average rate of change:
\[ \frac{g(7) - g(2)}{7 - 2} = \frac{-41 - 19}{7 - 2} = \frac{-60}{5} = -12 \]
So, the average rate of change of the function over the interval 2 < x < 7 is -12, which corresponds to answer D.