calculus ( differentiation of logarithm function)

Question

Consider the function: 𝑓(𝑥) = ln x /x a. Determine the intervals over which the function is increasing and over which it is decreasing. b. Determine the intervals over which the function is concave up and over which it is concave down. c. Identify any/all relative extrema and inflection points.

Answer 1

Use quotient rule to get f'(x) = (1-ln(x))/x²

f(x) increases whenever f'(x)>0; f(x) decreases whenever f'(x)<0

Find f ''(x) = d/dx(f'(x)) to find concavity

f(x) is concave up whenever f ''(x)>0

f(x) is concave down whenever f''(x)<0