`y = root(3)(x^2 + 1)` Sketch the curve by locating max/mins, asymptotes, points of inflection, etc.

Textbook Question

Chapter 4, Review - Problem 28 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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a) Asymptotes

Since the function has no undefined point, so it has no vertical asymptote.

For horizontal asymptotes check if at x`->+-oo` , the function behaves as a line y=mx+b,

Compute `lim_(x->+-oo)f(x)/x` to find m


Compute `lim_(x->+-oo)f(x)-mx` to find b,


Since the result is not a finite constant , so there is no horizontal asymptote at `+-oo`

b) Maxima/Minima



Let's find critical numbers by solving x, for y'=0

`(2x)/(3(x^2+1)^(2/3))=0 ,rArrx=0`  

Let's check the sign of y' by plugging test point in the intervals (-`oo` ,0) and (0,`oo` )



There is no maximum point.

Minimum point is at x=0

c) Inflection Points




Let's find inflection points by solving x, for y''=0

`3-x^2=0 , x=+-sqrt(3)`

Inflection points are (`sqrt(3)` ,`root(3)(4)` ) and (-`sqrt(3)` ,`root(3)(4)` )

Graph: Function is plotted in red color and second derivative is plotted in green color.

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