Chapter 3: Q. 75 (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 $f$ .
$f (x) = x \ln x$ .

Short Answer

Expert verified

The graph for the function $f (x) = x \ln x$ is,

Step by step solution

Step 1. Given information

$f (x) = x \ln x$ .

Step 2. Let src="data:image/svg+xml;base64,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" role="math" localid="1648548589163" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/e3a39aa2-cffb-4e88-9920-a390b0fd0716.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220329%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220329T101855Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=979c14ce910f9452023d7a198bae8b26913addddbfca498d82c264ef2af661e8" y=xln x.

Now point table for the function is given by,

$x$	$y$	$(x, y)$
$0.0001$	$0$	$(0.0001, 0)$
$1$	$0$	$(1, 0)$
$2$	$1.386$	$(2, 1.386)$
$3$	$3.296$	$(3, 3.296)$

Step 3. The graph for the function is,

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

$\frac{d}{d x} (x \ln x) = 0 x \cdot \frac{1}{x} + \ln x \cdot 1 = 0 1 + \ln x = 0 \ln x = - 1 x = e^{- 1}$

Therefore, $f$ has a critical point at $x = \frac{1}{e}$ . It has a local minima at $x = \frac{1}{e}$ .

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

For roots of the function,

$x \ln x = 0 \ln x = 0 x = e^{0} x = 1$

Step 6. Therefore, the function f is defined on 0,∞.

The function is positive on $(1, \infty)$ and negative elsewhere it is defined. The function have a local minimum at $x = e^{- 1}$ . The function is increasing on $(\frac{1}{e}, \infty)$ and decreasing elsewhere it is defined.

Again,

$\lim_{x \to 0^{+}} f (x) = \lim_{x \to 0^{+}} x \ln x = \infty \lim_{x \to \infty} f (x) = \lim_{x \to \infty} x \ln x = \infty$

$\lim_{x \to 0^{+}} f (x) = \infty$ So, there is a vertical asymptote at $x = 0$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Step 1. Given information

Step 3. The graph for the function is,

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

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

Step 6. Therefore, the function f is defined on 0,∞.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.