Question

select the correct answer.
consider this function.
$f(x) = 6\log_{2} x - 3$
over which interval is function $f$ increasing at the greatest rate?
\\(\bigcirc\\) a. \\(2, 6\\)
\\(\bigcirc\\) b. \\(\left\frac{1}{8}, \frac{1}{2}\
ight\\)
\\(\bigcirc\\) c. \\(1, 2\\)
\\(\bigcirc\\) d. \\(\left\frac{1}{2}, 1\
ight\\)

Explanation:

Step1: Find the derivative of \( f(x) \)

The function is \( f(x) = 6\log_2 x - 3 \). The derivative of \( \log_b x \) is \( \frac{1}{x\ln b} \), so the derivative of \( \log_2 x \) is \( \frac{1}{x\ln 2} \). Then the derivative of \( f(x) \) is \( f'(x)=\frac{6}{x\ln 2} \).

Step2: Analyze the derivative's behavior

The function \( f'(x)=\frac{6}{x\ln 2} \) is a positive function (since \( x>0 \) for the logarithm to be defined, and \( 6,\ln 2>0 \)) and it is a decreasing function of \( x \) (because it has the form \( \frac{k}{x} \) with \( k>0 \)). So, the smaller the value of \( x \) in the interval, the larger \( f'(x) \) is, meaning the function \( f(x) \) is increasing at a greater rate.

Step3: Compare the intervals

We look at the left - hand endpoints of each interval (since the derivative is decreasing, the left - most \( x \) in the interval will give the largest derivative in that interval):

• For interval A: \( [2,6] \), left - hand endpoint \( x = 2 \)
• For interval B: \( [\frac{1}{8},\frac{1}{2}] \), left - hand endpoint \( x=\frac{1}{8} \)
• For interval C: \( [1,2] \), left - hand endpoint \( x = 1 \)
• For interval D: \( [\frac{1}{2},1] \), left - hand endpoint \( x=\frac{1}{2} \)

Among \( \frac{1}{8}, \frac{1}{2},1,2 \), \( \frac{1}{8} \) is the smallest. So the interval \( [\frac{1}{8},\frac{1}{2}] \) has the smallest \( x \) values (on the left - hand side of the interval), so \( f'(x) \) will be largest in this interval, meaning \( f(x) \) is increasing at the greatest rate in this interval.

Answer:

B. \( [\frac{1}{8},\frac{1}{2}] \)