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

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Textbook Question

Chapter 4, Review - Problem 27 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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a) Asymptotes

Domain of function is x `>=` -2

Since the function has no undefined points , so it has no vertical asymptotes.

Horizontal Asymptotes:

Let's find the limits of the function at `+-oo`

Since -`oo` is not in the domain so there is no horizontal asymptote at -`oo`

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



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

b) Maxima/Minima





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


`3x+4=0 ,=>x=-4/3`

Let's check the sign of y' by plugging test points in the intervals (-2,-4/3) and (-4/3 ,`oo` )



There is no maxima.

Minimum point is at x=-4/3


c) Inflection point

Let's find the second derivative of the function by using quotient rule,





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


`3x+8=0 ,harrx=-8/3`

However x=-8/3 is not in the domain of the function , so there are no inflection points.


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