Q. 5 Sign analyses for second derivat... [FREE SOLUTION]

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Chapter 3: Q. 5 (page 250)

Sign analyses for second derivatives: Repeat the instructions of the previous block of problems, except find sign intervals for the second derivative $f''$ instead of the first derivative.
$f (x) = \frac{x}{x^{2} + 1}$

Short Answer

Expert verified

The second derivative of the given function is the local maximum at $x = 1,$ and a local minimum at $x = - 1 .$

Step by step solution

Step 1. Given Information.

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

Step 2. Find the second derivative.

To find the second derivative, first, we find the first derivative then the second.

So,

$f (x) = \frac{x}{(x^{2} + 1)} f' (x) = \frac{(x^{2} + 1) 鈭� 2 x^{2}}{{(x^{2} + 1)}^{2}} f' (x) = \frac{1}{(x^{2} + 1)} 鈭� \frac{2 x^{2}}{{(x^{2} + 1)}^{2}} f'' (x) = \frac{鈭� 2 x}{{(x^{2} + 1)}^{2}} 鈭� \frac{{(x^{2} + 1)}^{2} 4 x 鈭� 4 x^{2} (x^{2} + 1) 鈰� 2 x}{{(x^{2} + 1)}^{4}} f'' (x) = \frac{鈭� 2 x}{{(x^{2} + 1)}^{2}} 鈭� \frac{4 x (x^{2} + 1) [x^{2} + 1 鈭� 2 x^{2}]}{{(x^{2} + 1)}^{4}} f'' (x) = \frac{鈭� 2 x}{{(x^{2} + 1)}^{2}} 鈭� 4 x \frac{[鈭� x^{2} + 1]}{{(x^{2} + 1)}^{3}} f'' (x) = \frac{1}{{(x^{2} + 1)}^{3}} [鈭� 2 x^{3} 鈭� 2 x + 4 x^{3} 鈭� 4 x] f'' (x) = \frac{1}{{(x^{2} + 1)}^{3}} [鈭� 6 x + 2 x^{3}] f'' (x) = \frac{鈭� 2 x (3 + x^{2})}{{(x^{2} + 1)}^{3}}$

Step 3. Find sign intervals for the second derivative.

To find the sign intervals let's find the critical points from the first derivative.

So,

$f' (x) = \frac{(x^{2} + 1) - 2 x^{2}}{{(x^{2} + 1)}^{2}} 0 = \frac{(x^{2} + 1) - 2 x^{2}}{{(x^{2} + 1)}^{2}} 0 = (x^{2} + 1) - 2 x^{2} 0 = - x^{2} + 1 x = 卤 1$

Thus, the critical points are $x = 1 and x = - 1 .$

Now, at $x = 1, f'' (x) < 0$ and at $x = - 1, f'' (x) > 0 .$

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91影视

Sign analyses for second derivatives: Repeat the instructions of the previous block of problems, except find sign intervals for the second derivative $f''$ instead of the first derivative.
$f (x) = \frac{x}{x^{2} + 1}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the second derivative.

Step 3. Find sign intervals for the second derivative.

One App. One Place for Learning.

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91影视

Sign analyses for second derivatives: Repeat the instructions of the previous block of problems, except find sign intervals for the second derivative f''instead of the first derivative.fx=xx2+1

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the second derivative.

Step 3. Find sign intervals for the second derivative.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

Geometry

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.

Sign analyses for second derivatives: Repeat the instructions of the previous block of problems, except find sign intervals for the second derivative $f''$ instead of the first derivative.
$f (x) = \frac{x}{x^{2} + 1}$