Question
Over what intervals is the average rate of change of f(x) = 5x greater than the average rate of change of g(x) = 25x?
Answers
GPT-4o mini
To find the intervals where the average rate of change of \( f(x) = 5x \) is greater than the average rate of change of \( g(x) = 25x \), we begin by calculating the average rate of change for the functions \( f \) and \( g \) over an interval \([a, b]\).
The average rate of change of a function \( h(x) \) from \( a \) to \( b \) is given by:
\[
\text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a}
\]
Let's calculate the average rates of change for \( f(x) \) and \( g(x) \).
1. For \( f(x) = 5x \):
\[
\text{Average rate of change of } f(x) = \frac{f(b) - f(a)}{b - a} = \frac{5b - 5a}{b - a} = \frac{5(b - a)}{b - a} = 5
\]
2. For \( g(x) = 25x \):
\[
\text{Average rate of change of } g(x) = \frac{g(b) - g(a)}{b - a} = \frac{25b - 25a}{b - a} = \frac{25(b - a)}{b - a} = 25
\]
Now, we set up the inequality to find the intervals where the average rate of change of \( f \) is greater than that of \( g \):
\[
5 > 25
\]
This inequality is false; thus, there are no intervals where the average rate of change of \( f(x) \) is greater than that of \( g(x) \).
In conclusion, the average rate of change of \( f(x) = 5x \) is not greater than the average rate of change of \( g(x) = 25x \) for any intervals \( [a, b] \).
The average rate of change of a function \( h(x) \) from \( a \) to \( b \) is given by:
\[
\text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a}
\]
Let's calculate the average rates of change for \( f(x) \) and \( g(x) \).
1. For \( f(x) = 5x \):
\[
\text{Average rate of change of } f(x) = \frac{f(b) - f(a)}{b - a} = \frac{5b - 5a}{b - a} = \frac{5(b - a)}{b - a} = 5
\]
2. For \( g(x) = 25x \):
\[
\text{Average rate of change of } g(x) = \frac{g(b) - g(a)}{b - a} = \frac{25b - 25a}{b - a} = \frac{25(b - a)}{b - a} = 25
\]
Now, we set up the inequality to find the intervals where the average rate of change of \( f \) is greater than that of \( g \):
\[
5 > 25
\]
This inequality is false; thus, there are no intervals where the average rate of change of \( f(x) \) is greater than that of \( g(x) \).
In conclusion, the average rate of change of \( f(x) = 5x \) is not greater than the average rate of change of \( g(x) = 25x \) for any intervals \( [a, b] \).