Q.49 Use a sign chart for f''聽to det... [FREE SOLUTION]

Chapter 3: Q.49 (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
$f (x) = e^{1 + 2 x - x^{2}}$

Short Answer

Expert verified

Inflection point is $x = \frac{\sqrt{2} + 2}{2}, - \frac{\sqrt{2} - 2}{2}$ , concave up on $(- 鈭�, - \frac{\sqrt{2} - 2}{2}) (\frac{\sqrt{2} + 2}{2}, 鈭�)$ and concave down on $(- \frac{\sqrt{2} - 2}{2}, \frac{\sqrt{2} + 2}{2}) .$

Step by step solution

Step 1. Given Information.

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

Step 2. Second-derivative.

On differentiating,

$f^{'} (x) = \frac{d}{d x} e^{1 + 2 x - x^{2}} = e^{1 + 2 x - x^{2}} (2 - 2 x) f^{''} (x) = \frac{d}{d x} e^{1 + 2 x - x^{2}} (2 - 2 x) = 2 (2 x^{2} - 4 x + 1) e^{1 + 2 x - x^{2}}$

Step 3. Sign chart.

Now, $f'' (x) = 0 a t x = \frac{\sqrt{2} + 2}{2}, - \frac{\sqrt{2 - 2}}{2} .$

Therefore, these are the inflection points.

The sign chart will be:

Concave up on $(- 鈭�, - \frac{\sqrt{2} - 2}{2}) (\frac{\sqrt{2} + 2}{2}, 鈭�)$

Concave down on $(- \frac{\sqrt{2} - 2}{2}, \frac{\sqrt{2} + 2}{2})$

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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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)=e1+2x-x2

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

Theoretical and Mathematical Physics

Pure Maths

Calculus

Geometry

Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.