To determine which interval shows the greatest increase of the function [tex]\( f \)[/tex], we need to examine the average rates of change over each interval provided in the table. The average rate of change essentially indicates how much the function [tex]\( f \)[/tex] increases (or decreases) on average over that specific interval.
Here are the average rates of change for each interval:
- For [tex]\( 0 \leq x \leq 1 \)[/tex], the average rate of change is 10.
- For [tex]\( 1 \leq x \leq 4 \)[/tex], the average rate of change is -5.
- For [tex]\( 4 \leq x \leq 8 \)[/tex], the average rate of change is 2.
- For [tex]\( 8 \leq x \leq 10 \)[/tex], the average rate of change is 6.
To find out where the function [tex]\( f \)[/tex] increases the most, we need to identify the interval with the highest positive rate of change. Let's compare the given values:
- A rate of change of 10 (for [tex]\( 0 \leq x \leq 1 \)[/tex])
- A rate of change of -5 (for [tex]\( 1 \leq x \leq 4 \)[/tex])
- A rate of change of 2 (for [tex]\( 4 \leq x \leq 8 \)[/tex])
- A rate of change of 6 (for [tex]\( 8 \leq x \leq 10 \)[/tex])
Clearly, a rate of change of 10 is the highest among these values, and it's associated with the interval [tex]\( 0 \leq x \leq 1 \)[/tex]. Therefore, the function [tex]\( f \)[/tex] increases the most on the interval [tex]\( 0 \leq x \leq 1 \)[/tex].
So the answer is:
(A) [tex]\( 0 \leq x \leq 1 \)[/tex]
