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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Find the vertical and horizontal asymptotes.

(b) Find the intervals of increase or decrease.

(c) Find the local maximum and minimum values.

(d) Find the intervals of concavity and the inflection points.

(e) Use the information from parts (a)−(d) to sketch the graph of f.

f(x) = ln(1 − ln x)

(a)

To determine

To find: The vertical and horizontal asymptotes of the function f(x)=ln(1lnx) .

Explanation

Calculation:

Vertical asymptote is obtain at x=a , if limxaf(x)= or limxaf(x)= .

As x approaches 0,

limx0f(x)=limx0ln(1lnx)=ln(1limx0lnx)=ln(1)=

As x approaches e,

limxef(x)=limxeln(1lnx)=ln(1limxelnx)=ln(1lne)=ln(11)

Simplify further to check the vertical asymptote at x=e .

limxef(x)=ln(0)=

Hence, the function is undefined when x=0 and x=e

(b)

To determine

To find: On which intervals the function f(x)=ln(1lnx) is increasing or decreasing.

(c)

To determine

To find: The local maximum and minimum values of the function f(x)=ln(1lnx) .

(d)

To determine

To find: The intervals of concavity and the inflection points of the function f(x)=ln(1lnx) .

(e)

To determine

To sketch: The graph of the function for the above information from part (a) to part (d).

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