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Question 4

\begin{tabular}{|l|l|l|l|l|}
\hline
Interval & [tex]$0 \leq x \leq 1$[/tex] & [tex]$1 \leq x \leq 4$[/tex] & [tex]$4 \leq x \leq 8$[/tex] & [tex]$8 \leq x \leq 10$[/tex] \\
\hline
Average Rate of Change & 10 & -5 & 2 & 6 \\
\hline
\end{tabular}

The table gives the average rates of change of a function [tex]$f$[/tex] over different intervals. On which of the intervals does the function increase the most?

A. [tex]$0 \leq x \leq 1$[/tex]
B. [tex]$1 \leq x \leq 4$[/tex]
C. [tex]$4 \leq x \leq 8$[/tex]
D. [tex]$8 \leq x \leq 10$[/tex]

Sagot :

GinnyAnsweridn
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]
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