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?
A. (2,6)
B. (1,2)
C. ((1)/(2),1)
D.((1)/(8),(1)/(2))

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine the interval over which the function is increasing at the greatest rate, we need to find the interval where the derivative of the function is the largest.

To find the derivative of the function, we can use the chain rule.

f'(x) = (6/(ln(2)))(1/x)

Now, we need to determine which interval will result in the largest value for f'(x).

To do this, we can find the critical points of f'(x) by setting the derivative equal to zero and solving for x.

(6/(ln(2)))(1/x) = 0

This equation has no solution since a fraction can only equal zero if its numerator is zero, and 6/(ln(2)) is not equal to zero.

Therefore, there are no critical points for f'(x), and the function is always increasing or always decreasing.

Since we are looking for the interval where the function is increasing at the greatest rate, we can conclude that the function is increasing at the same rate over its entire domain.

Therefore, the answer is:

D. ((1)/(8),(1)/(2))