Final answer:
The function f(x) = 4x - 12ln(6x) has a critical point at x = 0.5, where it has a local maximum. The function is increasing on the interval (0, 0.5) and decreasing on the interval (0.5, ∞).
Step-by-step explanation:
To find the critical point and the intervals on which the function f(x) = 4x - 12ln(6x), where x > 0, is increasing or decreasing, we must first find the first derivative of the function.
The derivative of f(x) is f'(x) = 4 - 12/(6x). Simplifying this, we get f'(x) = 4 - 2/x. To find the critical points, we set the derivative equal to zero: 4 - 2/x = 0, which simplifies to x = 0.5.
To apply the First Derivative Test, we determine the sign of the first derivative on either side of the critical point x = 0.5. If f'(x) is positive to the left of 0.5 and negative to the right, the function increases to the left of 0.5 and decreases to the right, indicating a local maximum at x = 0.5. Conversely, if f'(x) is negative to the left and positive to the right, x = 0.5 would be a local minimum.
Let's evaluate the sign of f' on either side of x = 0.5:
For x < 0.5, say x = 0.25, f'(0.25) = 4 - 2/0.25 = 4 - 8 = -4, which is negative.
For x > 0.5, say x = 1, f'(1) = 4 - 2/1 = 2, which is positive.
Hence, f(x) has a critical point at x = 0.5 where the function transitions from increasing to decreasing, and this point is a local maximum.