   Chapter 4.3, Problem 51E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### 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 ) = x 2 + 1 − x

(a)

To determine

To find: The vertical and horizontal asymptotes of the function f(x)=x2+1x .

Explanation

Calculation:

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

The given function is defined for all x. It cannot be undefined. Therefore,

the function has no vertical asymptote.

Horizontal asymptote is obtain if limx±f(x)=k where k is any real number.

When x approaches ,

limxf(x)=limx(x2+1x)=limx(x2+1x×x2+1+xx2+1+x)=limx((x2+1)2(x)2x2+1+x)=limx(x2+1x2x2+1+x)

On further simplification find the horizontal asymptote

(b)

To determine

To find: On which intervals the function f(x)=x2+1x is increasing or decreasing.

(c)

To determine

To find: The local maximum and minimum values of the given function f(x)=x2+1x .

(d)

To determine

To find: The intervals of concavity and the inflection points of the function f(x)=x2+1x .

(e)

To determine

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

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