Determine the intervals on which the function f(x) = 6x - 9 ln (4x), (x > 0) is

Question

Determine the intervals on which the function f(x) = 6x - 9 ln (4x), (x > 0) is concave up or down and find the points of inflection (Use symbolic notation and fractions where needed. Give your answer as comma separated list. Enter NULL if there are no points of infection) Points of inflection: (Use symbolic notation and fractions where needed. Give your answers as intervals in the form (* middot *). Use inf for infinity. U for combining intervals, and appropriate type parallel depending on whether the interval is open or closed. Enter NULL if interval is empty.) f(x) is concave up when x epsilon f(x) is concave down when x epsilon

Explanation / Answer

Given function

f(x) = 6x - 9 ln (x)

find first derivative

f'(x) = 6 - 9(1/x)

find second derivative

f"(x) = - 9 (-1) (x-2)

= 9/x^2

set f"(x) = 0

which gives 9/x^2=0

x = not defined

so it can be said that point of inflection x = [ -inf, +inf]

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