Chapter 3: Q. 51 (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
$f (x) = \sin (x - \frac{π}{4})$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

Step 2. Second derivative.

On differentiating the function, we get,

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

Step 3. Sign chart.

Now,

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

Therefore, the chart will look like,

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

Step 4. Verification.

The graph of the function is,

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Second derivative.

Step 3. Sign chart.

Step 4. Verification.

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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 utilityf(x)=sinx-π4

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

Mechanics Maths

Calculus

Geometry

Probability and Statistics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.