Q. 69 - Calculus

Question

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$ and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=\sqrt{x^2+1}$.

Step-by-Step Solution

Answer

The graph for the function $f\left(x\right)=\sqrt{x^2+1}$ is,

Step 1 . Given information

$f\left(x\right)=\sqrt{x^2+1}$.

Step 2 . Let y = x 2 + 1 .

Now point table for the function is given by,

$x$	$y$	$\left(x,y\right)$
$-2$	$\sqrt{5}$	$\left(-2,\sqrt{5}\right)$
$-1$	$\sqrt{2}$	$\left(-1,\sqrt{2}\right)$
$0$	$1$	$\left(0,1\right)$
$1$	$\sqrt{2}$	$\left(1,\sqrt{2}\right)$
$2$	$\sqrt{5}$	$\left(2,\sqrt{5}\right)$

Step 3 . The graph for the function is,

Step 4 . Now for critical point f ' x = 0 .

$$\begin{gathered} \frac{d}{dx}\left(\sqrt{x^2+1}\right)=0 \\ \frac{1}{2\sqrt{x^2+1}}.2x=0 \\ \frac{x}{\sqrt{x^2+1}}=0 \\ x=0 \end{gathered}$$

Therefore, $f$ has a critical point at $x=0$. So, it has a local minimum at $x=0$.

Step 5 . The sign chart of f is shown below:

For roots of the function,

$$\begin{gathered} \sqrt{x^2+1}=0 \\ {x}^2+1=0 \\ {x}^2=-1 \end{gathered}$$

Therefore, $x$ has no real value.

Step 6 . Therefore, the function f is increasing on 0 , ∞ and decreasing on - ∞ , 0 .

Again,

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

Therefore, the function is defined everywhere and it has no root. Positive everywhere.