Question

Select the correct answer.

Consider this function.
f (x) = 6log2 x-3
Over which interval is function f increasing at the greatest rate?
A. [1/8, 1/2]
B. [2,6]
C. [1,2]
D. [1/2,1]

Answer 1

The derivative of f(x) is f'(x) = 6/(x ln 2).

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

Since the derivative is always positive, we just need to find the interval where the derivative is the largest. This occurs when x is the smallest.

So, we need to find the smallest value of x in each interval, and then compare the corresponding values of the derivative.

For interval [1/8, 1/2]:

The smallest value of x is 1/8, so f'(1/8) = 6/(1/8 ln 2) = 48 ln 2.

For interval [2,6]:

The smallest value of x is 2, so f'(2) = 6/(2 ln 2) = 3 ln 2.

For interval [1,2]:

The smallest value of x is 1, so f'(1) = 6/(1 ln 2) = 6 ln 2.

For interval [1/2,1]:

The smallest value of x is 1/2, so f'(1/2) = 6/(1/2 ln 2) = 12 ln 2.

Comparing these values, we can see that the largest value is 48 ln 2, which occurs in interval [1/8, 1/2].

Therefore, the answer is A. [1/8, 1/2].