f(x)=\\sqrt{x^{2}+2}-x\n(a) find the vertical asymptote(s). (enter your answers as a comma - separated…

Explanation:

Step1: Analyze vertical asymptote

For a function (y = f(x)=\sqrt{x^{2}+2}-x), the domain is all real numbers ((x\in R)) since (\sqrt{x^{2}+2}) is defined for all (x). A vertical asymptote occurs when the function approaches (\pm\infty) as (x) approaches a certain value. But (\lim_{x\rightarrow a}(\sqrt{x^{2}+2}-x)) is finite for all (a\in R). So, there is no vertical asymptote.

Step2: Analyze horizontal asymptote

We find (\lim_{x\rightarrow\infty}(\sqrt{x^{2}+2}-x)). Multiply by (\frac{\sqrt{x^{2}+2}+x}{\sqrt{x^{2}+2}+x}): [ \begin{align*} \lim_{x\rightarrow\infty}(\sqrt{x^{2}+2}-x)&=\lim_{x\rightarrow\infty}\frac{(\sqrt{x^{2}+2}-x)(\sqrt{x^{2}+2}+x)}{\sqrt{x^{2}+2}+x}\ &=\lim_{x\rightarrow\infty}\frac{x^{2}+ 2-x^{2}}{\sqrt{x^{2}+2}+x}\ &=\lim_{x\rightarrow\infty}\frac{2}{\sqrt{x^{2}+2}+x}\ & = 0 \end{align*} ] Also, (\lim_{x\rightarrow-\infty}(\sqrt{x^{2}+2}-x)=\lim_{x\rightarrow-\infty}\vert x\vert\sqrt{1+\frac{2}{x^{2}}}-x). Since (x\rightarrow-\infty), (\vert x\vert=-x), then (\lim_{x\rightarrow-\infty}(-x\sqrt{1+\frac{2}{x^{2}}}-x)=\lim_{x\rightarrow-\infty}-x\left(\sqrt{1+\frac{2}{x^{2}}}+1\right)=\infty)

Step3: Find the derivative for monotonicity

First, find the derivative (y'=\frac{x}{\sqrt{x^{2}+2}}-1=\frac{x-\sqrt{x^{2}+2}}{\sqrt{x^{2}+2}}). Since (x-\sqrt{x^{2}+2}<0) for all (x\in R) (because (\sqrt{x^{2}+2}>|x|\geq x)), (y'<0) for all (x\in R)

Step4: Analyze local extrema

Since the function is always decreasing ((y'<0) for all (x\in R)), there are no local maximum or local minimum values

Step5: Find the second - derivative for concavity

Find the second - derivative. (y'=\frac{x}{\sqrt{x^{2}+2}}-1), then (y''=\frac{\sqrt{x^{2}+2}-\frac{x^{2}}{\sqrt{x^{2}+2}}}{x^{2}+2}=\frac{2}{(x^{2}+2)^{\frac{3}{2}}}>0) for all (x\in R)

Answer:

(a) (x=\text{None}), (y = 0) (b) Interval of increase: (\text{None}), Interval of decrease: ((-\infty,\infty)) (c) Local minimum value(s): (\text{None}), Local maximum value(s): (\text{None}) (d) Inflection point: (\text{None}), Interval where the graph is concave upward: ((-\infty,\infty)), Interval where the graph is concave downward: (\text{None})