💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!



Numerade Educator



Problem 51 Hard Difficulty

(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) = \sqrt{x^2 + 1} - x $


(a) $\lim _{x \rightarrow-\infty}(\sqrt{x^{2}+1}-x)=\infty$ and
$\lim _{x \rightarrow \infty}(\sqrt{x^{2}+1}-x)=\lim _{x \rightarrow \infty}(\sqrt{x^{2}+1}-x) \frac{\sqrt{x^{2}+1}+x}{\sqrt{x^{2}+1}+x}=\lim _{x \rightarrow \infty} \frac{1}{\sqrt{x^{2}+1}+x}=0,$ so $y=0$ is a HA
(b) $f(x)=\sqrt{x^{2}+1}-x \Rightarrow f^{\prime}(x)=\frac{x}{\sqrt{x^{2}+1}}-1 .$ since $\frac{x}{\sqrt{x^{2}+1}}<1$ for all $x, f^{\prime}(x)<0,$ so $f$ is decreasing on $\mathbb{R}$
(c) No minimum or maximum
(d) $f^{\prime \prime}(x)=\frac{\left(x^{2}+1\right)^{1 / 2}(1)-x \cdot \frac{1}{2}\left(x^{2}+1\right)^{-1 / 2}(2 x)}{(\sqrt{x^{2}+1})^{2}}$
$=\frac{\left(x^{2}+1\right)^{1 / 2}-\frac{x^{2}}{\left(x^{2}+1\right)^{1 / 2}}}{x^{2}+1}=\frac{\left(x^{2}+1\right)-x^{2}}{\left(x^{2}+1\right)^{3 / 2}}=\frac{1}{\left(x^{2}+1\right)^{3 / 2}}>0$


You must be signed in to discuss.

Video Transcript

The problem is part of a find the vertical and horizontal Azem totes we'LL put a so we have half works is equal to x squared plus one square root of X squared plus one bus Thanks Hams scored a tiff Ex Squire Paswan minus X over square root of X squared plus Juan US max which is equal to one over square root of X squared plus one us max. So we're half the limit. Ex gustatory infinity for box is equal to you Cyril No one acts goes to negative infinity from the function With half after wax goes through infinity over half Why come to zero is horizontal Azem tote And there is no work ical Azem told off this function had to be find the intervals of increase or decrease first of computers Derivative This is Echo two Act's over square root of X squared plus one minus one Not his side Axe is last time Explore your past Come on So it's a generative is smaller than Siro We'Ll all relax So we have I've is in decreasing home Negative infinity to infinity How to see find it a local maximum and minimum values from hot be We know there's no local maximum on minimal values Hot tea Find the intervals off Comm cavity onda the inflection points So we're half second. Derivative is the control square root of X squared plus one minus X times one half times Truax times x squared plus one The power ofthe negative behalf over X squared plus one which is equal to one over X square plus one. It was a pop off three over too, which is always grazes and Cyril. So we're half the function after Lex is Can Kev upward negative infinity to infinity. So seriously, no inflection point how to eat Using the information from parts A to B to scratch the graph of us so we can sketch the graph as follows This is a graph of life.