Chapter 3: Q. 53 (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
$f (x) = 3 \cos (\frac{π}{2} x) + 5$

Short Answer

Expert verified

Inflection points are $x = 1 + 4 n, 3 + 4 n,$ concave up on $(1 + 4 n, 3 + 4 n)$ and concave down on $(4 n, 1 + 4 n), (3 + 4 n, 4 + 4 n) .$

Step by step solution

Sep 1. Given information.

The given function is $f (x) = 3 \cos (\frac{π}{2} x) + 5 .$

Step 2. Second derivative.

On differentiating, we get,

$f^{'} (x) = \frac{d}{d x} (3 \cos (\frac{π}{2} x) + 5) = - \frac{3 π}{2} \sin (\frac{π}{2} x) f^{''} (x) = \frac{d}{d x} (- \frac{3 π}{2} \sin (\frac{π}{2} x)) = - \frac{3 π^{2}}{4} \cos (\frac{π}{2} x)$

Step 3. Sign chart.

Now,

$f'' (x) = 0 a t x = 1 + 4 n, 3 + 4 n .$

Inflection points: $x = 1 + 4 n, 3 + 4 n .$

Concave up: $(1 + 4 n, 3 + 4 n)$

Concave down: $(4 n, 1 + 4 n), (3 + 4 n, 4 + 4 n)$

Step 4. Verification.

The graph of the function is :

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Short Answer

Step by step solution

Sep 1. Given information.

Step 2. Second derivative.

Step 3. Sign chart.

Step 4. Verification.

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Use a sign chart forf'' 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)=3cosπ2x+5

Short Answer

Step by step solution

Sep 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

Discrete Mathematics

Calculus

Theoretical and Mathematical Physics

Mechanics Maths

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.