Chapter 3: Q. 21 (page 314)

For each function f that follows, construct sign charts forf, f',and f '', if possible. Examine function values or limits at any interesting values and at ±∞. Then interpret this information to sketch a labeled graph of f.
$f (x) = x e^{x} .$

Short Answer

Expert verified

Sign charts forf, f',and f ''are the following.

Graph of function is following.

Step by step solution

Step 1. Given information.

The given function is $f (x) = x e^{x} .$

Step 2. Roots of the function.

Equate the function to zero.

$f (x) = 0 x e^{x} = 0 x = 0$

the function root of the function is $x = 0 .$

Step 3. critical points of the function.

Determine the first derivative of the function.

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

critical points of the function.

$f' (x) = 0 (x + 1) e^{x} = 0 x + 1 = 0 x = - 1$

so the function has critical point at $x = - 1 .$

Step 4. inflection point of the function.

find the second derivative of the function.

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

Roots of the second derivative of the function.

$f'' (x) = 0 (x + 2) e^{x} = 0 x + 2 = 0 x = - 2$

So the function has an inflection point at $x = - 2 .$

So the sign chart is the following.

Step 5. Graph of function.

The graph of the function is the following.

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For each function f that follows, construct sign charts forf, f',and f '', if possible. Examine function values or limits at any interesting values and at ±∞. Then interpret this information to sketch a labeled graph of f.
$f (x) = x e^{x} .$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Roots of the function.

Step 3. critical points of the function.

Step 4. inflection point of the function.

Step 5. Graph of function.

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For each function f that follows, construct sign charts forf, f',and f '', if possible. Examine function values or limits at any interesting values and at ±∞. Then interpret this information to sketch a labeled graph of f.f(x)=xex.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Roots of the function.

Step 3. critical points of the function.

Step 4. inflection point of the function.

Step 5. Graph of function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Discrete Mathematics

Mechanics Maths

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

For each function f that follows, construct sign charts forf, f',and f '', if possible. Examine function values or limits at any interesting values and at ±∞. Then interpret this information to sketch a labeled graph of f.
$f (x) = x e^{x} .$