Select the correct answer.
Consider this function.
[tex]$f(x)=6 \log _2 x-3$[/tex]
Over which interval is function [tex]$f$[/tex] increasing at the greatest rate?
A. [tex]$\left[\frac{1}{8}, \frac{1}{2}\right]$[/tex]
B. [tex]$\left[\frac{1}{2}, 1\right]$[/tex]
C. [tex]$[2,6]$[/tex]
D. [tex]$[1,2]$[/tex]
Answer (1)
Find the derivative of the function f ( x ) = 6 lo g 2 x − 3 , which is f ′ ( x ) = x l n 2 6 .
Determine that the function increases at the greatest rate where the derivative is largest, which occurs when x is smallest.
Compare the smallest values in each interval: 8 1 , 2 1 , 2 , and 1 .
Conclude that the function increases at the greatest rate over the interval [ 8 1 , 2 1 ] .
Explanation
Find the derivative We are given the function f ( x ) = 6 lo g 2 x − 3 and asked to find the interval over which the function is increasing at the greatest rate. The rate of increase of a function is determined by its derivative. So, we need to find the derivative of f ( x ) .
Calculate the derivative The derivative of f ( x ) = 6 lo g 2 x − 3 is given by:
f ′ ( x ) = d x d ( 6 lo g 2 x − 3 ) = 6 ⋅ x l n 2 1 = x l n 2 6
The derivative f ′ ( x ) represents the rate of change of the function f ( x ) .
Find where the derivative is largest To find the interval where the function is increasing at the greatest rate, we need to find where the derivative f ′ ( x ) is largest. Since f ′ ( x ) = x l n 2 6 , the derivative is largest when x is smallest. We need to compare the given intervals to see which one contains the smallest values of x .
Compare the intervals The given intervals are:
A. [ 8 1 , 2 1 ] B. [ 2 1 , 1 ] C. [ 2 , 6 ] D. [ 1 , 2 ]
We need to find the smallest value of x within each interval. The smallest values are:
A. 8 1 = 0.125 B. 2 1 = 0.5 C. 2 D. 1
The smallest of these values is 8 1 .
Conclusion Since the smallest value of x is 8 1 , the function is increasing at the greatest rate over the interval [ 8 1 , 2 1 ] .
Examples
Imagine you are designing a water slide. The rate at which a slider's height changes determines the thrill. Similarly, understanding where a function increases most rapidly helps optimize designs for maximum effect, whether it's a slide, a roller coaster, or even the efficiency of an engine. By finding the interval where the function increases the most, you can optimize the design for the most thrilling experience.
