Q. 51 Use a sign chart for f''聽to det... [FREE SOLUTION]

Chapter 3: Q. 51 (page 275)

Use a sign chart for $f''$ to determine the intervals on which each function f in Exercises 41鈥�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
$f (x) = \sin (x - \frac{蟺}{4})$

Short Answer

Expert verified

Inflection points are at $x = \frac{蟺}{4} + 2 蟺 k, \frac{5 蟺}{4} + 2 蟺 k$ , concave up at all other places other than where it is concave down that is at $(\frac{蟺}{4} + 2 蟺 k, \frac{5 蟺}{4} + 2 蟺 k) .$

Step by step solution

Step 1. Given information.

The given function is $f (x) = \sin (x - \frac{蟺}{4}) .$

Step 2. Second derivative.

On differentiating the function, we get,

$f^{'} (x) = \frac{d}{d x} \sin (x - \frac{蟺}{4}) = \cos (x - \frac{蟺}{4}) f^{''} (x) = \frac{d}{d x} \cos (x - \frac{蟺}{4}) = - \sin (x - \frac{蟺}{4})$

Step 3. Sign chart.

Now,

$f'' (x) = 0 a t x = \frac{蟺}{4} + 2 蟺 k, \frac{5 蟺}{4} + 2 蟺 k$

Therefore, the chart will look like,

Inflection point is $x = \frac{蟺}{4} + 2 蟺 k, \frac{5 蟺}{4} + 2 蟺 k$ and concave up at all other places other than where it is concave down that is at $(\frac{蟺}{4} + 2 蟺 k, \frac{5 蟺}{4} + 2 蟺 k) .$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Pure Maths

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

91影视

Use a sign chart for f''to determine the intervals on which each function f in Exercises 41鈥�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 utilityf(x)=sinx-蟺4

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

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Pure Maths

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.