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(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) $f^{\prime \prime}(x)=\frac{\left(x^{2}+1\right)^{1 / 2}(1)-x \cdot \frac{1}{2}\left(x^{2}+1\right)^{-1 / 2}(2 x)}{(\sqrt{x^{2}+1})^{2}}$

$=\frac{\left(x^{2}+1\right)^{1 / 2}-\frac{x^{2}}{\left(x^{2}+1\right)^{1 / 2}}}{x^{2}+1}=\frac{\left(x^{2}+1\right)-x^{2}}{\left(x^{2}+1\right)^{3 / 2}}=\frac{1}{\left(x^{2}+1\right)^{3 / 2}}>0$

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