Chapter 3: Q. 50 (page 275)

Use a sign chart for $f''$ to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility
$f (x) = \frac{2^{x}}{1 - 2^{x}}$

Short Answer

Expert verified

There is no Inflection point . The function is concave up on $(- \infty, 0) and concave down on (0, \infty) .$

Step by step solution

Step 1. Given information.

The given function is $f (x) = \frac{2^{x}}{1 - 2^{x}}$ .

Step 2. Second Derivative.

On differentiating the function, we get,

$f^{'} (x) = \frac{d}{d x} \frac{2^{x}}{1 - 2^{x}} = \frac{(2^{x} \ln 2) (1 - 2^{x}) - (2^{x}) (\frac{d}{d x} 1 - \frac{d}{d x} 2^{x})}{{(1 - 2^{x})}^{2}} = \frac{2^{x} \ln 2}{{(2^{x} - 1)}^{2}} f^{''} (x) = \frac{d}{d x} \frac{2^{x} \ln 2}{{(2^{x} - 1)}^{2}} = - \frac{\ln^{2} (2) 2^{x} (2^{x} + 1)}{{(2^{x} - 1)}^{3}}$

Step 3. Sign chart.

Now,

$f'' (x) = 0 a t x = 0 .$

Therefore,

the sign chart will be,

The function is $concave up on (- \infty, 0) and concave down on (0, \infty)$ and no inflection point as domain is $(- \infty, 0) \cup (0, \infty) .$

Step 4. Verification.

The graph of the function is,

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 2. Second Derivative.

Step 3. Sign chart.

Step 4. Verification.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use a sign chart for f''to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility f(x)=2x1-2x

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Second Derivative.

Step 3. Sign chart.

Step 4. Verification.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.