Chapter 4.3, Problem 35E

(a)

To determine

To find : The vertical and horizontal asymptotes.

Explanation of Solution

Given: The function is $f\left(x\right)=\sqrt{x^2+1}-x$

Consider the function.

  $f\left(x\right)=\sqrt{x^2+1}-x$

The horizontal asymptote is calculated as,

  $\begin{array}{c}\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(\sqrt{x^2+1}-x\right)\\ =0\end{array}$

So, the horizontal asymptote is $f\left(x\right)=0$ .

The function has not undefined points so, there is no vertical asymptotes.

(b)

To determine

To find : The interval of increase or decrease.

Explanation of Solution

Given: The function is $f\left(x\right)=\sqrt{x^2+1}-x$

Consider the function.

  $f\left(x\right)=\sqrt{x^2+1}-x$

Differentiate the above expression with respect to $x$ .

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{1}{2\sqrt{x^2+1}}\left(2x\right)-1\\ =\frac{x}{\sqrt{x^2+1}}-1\end{array}$

The function is decreasing in the interval $\left(-\infty ,\infty \right)$

(c)

To determine

To find : The local maximum and minimum values..

Explanation of Solution

Given: The function is $f\left(x\right)=\sqrt{x^2+1}-x$

The function is decreasing in the interval $\left(-\infty ,\infty \right)$

There are no critical points for the function as the function is always decreasing.

Thus, there are no local maxima and minima points

(d)

To determine

To find : The intervals of concavity and the inflection points.

Explanation of Solution

Given: The function is $f\left(x\right)=\sqrt{x^2+1}-x$

Differentiate $f^{\prime }\left(x\right)$ with respect to $x$ .

  $\begin{array}{c}{f}^{″}\left(x\right)=\frac{\sqrt{x^2+1}-\frac{x^2}{\sqrt{x^2+1}}}{{\left(\sqrt{x^2+1}\right)}^2}\\ =\frac{x^2+1-x^2}{\left(x^2+1\right)\sqrt{x^2+1}}\\ =\frac{1}{{\left(x^2+1\right)}^{3/2}}\end{array}$

The value of $f^{″}\left(x\right)$ is .positive in for $-\infty <x<\infty$ So the concavity is upward in the interval $\left(-\infty ,\infty \right)$ .

The function has no inflection point.

(e)

To determine

To sketch : The graph of the function.

Explanation of Solution

Given: The function is $f\left(x\right)=\sqrt{x^2+1}-x$

Consider the function.

  $f\left(x\right)=\sqrt{x^2+1}-x$

The graph of the above function is shown in figure below.

  

Figure (1)

Therefore, the graph of the function is shown in Figure (1).