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Chapter 3: Q. 66 (page 261)

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 $f^{'}$ and examine any relevant limits so that you can describe all key points and behaviors of $f$ .
$f (x) = \sqrt{x} - \frac{1}{\sqrt{x}}$ .

Short Answer

Expert verified

The graph for the function $f (x) = \sqrt{x} - \frac{1}{\sqrt{x}}$ is,

Step by step solution

01

Step 1. Given information

$f (x) = \sqrt{x} - \frac{1}{\sqrt{x}}$ .

02

Step 2. Let y=x-1x.

Now point the table for the function is given by,

$x$	$y$	$(x, y)$
$1$	$0$	$(1, 0)$
$2$	$0.4142$	$(2, 0.4142)$
$4$	$1.5$	$(4, 1.5)$

03

Step 3. The graph for the function is,

04

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

$\frac{d}{d x} (\sqrt{x} - \frac{1}{\sqrt{x}}) = 0 \frac{d}{d x} (\frac{x - 1}{\sqrt{x}}) = 0 \frac{\sqrt{x} 路 \frac{1}{2 \sqrt{x}} - (x - 1) \frac{1}{2 \sqrt{x}}}{(\sqrt{x})^{2}} = 0 \frac{\frac{1}{2} - (x - 1) \frac{1}{2 \sqrt{x}}}{x} = 0 \frac{1 - x + 1}{2 x \sqrt{x}} = 0 \frac{x}{2 x \sqrt{x}} = 0 x = 0$

Therefore, $f$ has a critical point at $x = 0$ . But that is a local minimum at $x = 0$ .

05

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

For roots of the function,

$\sqrt{x} - \frac{1}{\sqrt{x}} = 0 \frac{x - 1}{\sqrt{x}} = 0 x - 1 = 0 x = 1$

06

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

Again,

$\lim_{x 鈫� - 鈭�} f (x) = \lim_{x 鈫� - 鈭�} (\sqrt{x} - \frac{1}{\sqrt{x}})$

$=$ Doesn't exist.

$\lim_{x 鈫� 鈭�} f (x) = \lim_{x 鈫� 鈭�} (\sqrt{x} - \frac{1}{\sqrt{x}}) = 鈭�$

Therefore, the function is defined on $(0, 鈭�)$ and the root at $x = - 1$ . Positive on $(1, 鈭�)$ and doesn't exist at $(- 鈭�, 0)$ .

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Most popular questions from this chapter

Find the possibility graph of its derivative f'.

Determine the graph of a function f from the graph of its derivative f'.

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

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

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points .

$f (x) = \frac{I n 2 x}{x}$

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

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