Question

Show that the function ln(x)/x is decreasing for x ≥?
A. e
B. π
C. 1
D. 0

Answer 1

To show that the function ln(x)/x is decreasing for x ≥ e, use the first derivative test. Find the first derivative of the function, set it equal to zero, and determine where it is negative. The function ln(x)/x is decreasing for x ≥ e. The correct answer is A. e.

Step-by-step explanation:

To show that the function ln(x)/x is decreasing for x ≥ e, we can use the first derivative test. Let's find the first derivative of the function and determine where it is negative.

Step 1: Find the first derivative of the function f(x) = ln(x)/x.
f'(x) = (1 - ln(x))/x^2

Step 2: Set the first derivative equal to zero and solve for x.
(1 - ln(x))/x^2 = 0
1 - ln(x) = 0
ln(x) = 1
x = e

Step 3: Determine where the first derivative is negative.
Since the first derivative f'(x) = (1 - ln(x))/x^2 is negative for x ≥ e, we can conclude that the function ln(x)/x is decreasing for x ≥ e.