Consider the following function. (If an answer does not exist, enter DNE.) f(x)

Question

Consider the following function. (If an answer does not exist, enter DNE.) f(x) In(3 In(x)) (a) Find the vertical asymptote(s). (Enter your answers as a comma separated list.) Find the horizontal asymptote(s). (Enter your answers as a comma-separated list.) (b) Find the interval where the function is increasing. (Enter your answer using interval notation.) Find the interval where the function is decreasing. (Enter your answer using interval notation.) (c) Find the local maximum and minimum values. local maximum value local minimum value (d) Find the interval where the function is concave up. (Enter your answer using interval notation.) Find the interval where the function is concave down (Enter your answer using interval notation.)

Explanation / Answer

From the given question,

f(x)=ln(3-ln (x))

(a) for vertical asymptote f(x) = infinity

3-ln (x)=0

3= ln (x)

x= e3

For horizontal asymptote,

as x tends to infinity, f(x)= ln( negative) , so f(x) doesnt exists.

No horizontal asymptote.

(b)f(x)=ln(3-ln (x))

f '(x)= 1/ (3-ln(x)) (-1/x)= -1/(x)(3-ln x)

for increasing function f '(x) >0

-1/(x)(3-ln x) >0

x>e3

for decreasing function f '(x) <0

-1/(x)(3-ln x) <0

0<x<e3

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