Final Answer:
The function is neither increasing nor decreasing over the given interval.
Step-by-step explanation:
To determine whether the function is increasing, decreasing, or neither over the given interval, we need to analyze the signs of the function’s derivative at each point in the interval. The derivative of the function is given by:
f’(x) = -2x + 4
We can evaluate the derivative at the points in the interval as follows:
f’(1) = -2(1) + 4 = -2 + 4 = 2 f’(4) = -2(4) + 4 = -8 + 4 = -4 f’(6) = -2(6) + 4 = -12 + 4 = -8 f’(8) = -2(8) + 4 = -16 + 4 = -12
Since the derivative is positive at points (1, -3) and (6, -7), and negative at points (4, -11) and (8, -10), the function is neither increasing nor decreasing over the interval. This is because the function has both positive and negative slopes over the interval, which means that it is not consistently increasing or decreasing.
To visualize this, we can graph the function on a coordinate plane using the given points. The graph will be a curve that connects the points (1, -3), (4, -11), (6, -7), and (8, -10). We can also draw lines connecting the points (8, -10), (10, -9), and (14, -9) to see that the function is not increasing or decreasing over the interval.