Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

Answers without the blur.

Just sign up for free and you're in.

Short Answer

In Exercises 83–86, use the given derivative $f'$ to find any local extrema and inflection points of f and sketch a possible graph without first finding a formula for f.
$f' (x) = x^{3} - 3 x^{2} + 3 x$

The possible graph of f is

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information.

The given derivative function is $f' (x) = x^{3} - 3 x^{2} + 3 x .$

Step 2. Apply the first derivative test.

To find the local extrema the first derivative of the function must be zero.

So,

$f' (x) = x^{3} - 3 x^{2} + 3 x 0 = x^{3} - 3 x^{2} + 3 x 0 = x (x^{2} - 3 x + 3) x = 0$

Thus, by the first derivative test, f has a local minimum at $x = 0 .$ The function has no local maxima.

Step 3. Finding inflection points.

An inflection point occurs when $f'' (x) = 0 .$

To find the inflection points, use the second derivative test.

So,

$f'' (x) = 3 x^{2} - 6 x + 3 0 = 3 x^{2} - 6 x + 3 0 = 3 (x^{2} - 2 x + 1) 0 = (x - 1) (3 x - 3) x = 1$

Thus, the function has an inflection point at $x = 1 .$ It is positive everywhere. Hence the graph of f will be concave up everywhere.

Step 4. Sketch the graph of function f.

The graph of the function is

Find Study Materials for

Create Study Materials

Select your language

Calculus

Answers without the blur.

Short Answer

In Exercises 83–86, use the given derivative $f'$ to find any local extrema and inflection points of f and sketch a possible graph without first finding a formula for f.
$f' (x) = x^{3} - 3 x^{2} + 3 x$

Step by Step Solution

Step 1. Given Information.

Step 2. Apply the first derivative test.

Step 3. Finding inflection points.

Step 4. Sketch the graph of function f.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Calculus

Answers without the blur.

Short Answer

In Exercises 83–86, use the given derivative f' to find any local extrema and inflection points of f and sketch a possible graph without first finding a formula for f.f'x=x3-3x2+3x

Step by Step Solution

Step 1. Given Information.

Step 2. Apply the first derivative test.

Step 3. Finding inflection points.

Step 4. Sketch the graph of function f.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

In Exercises 83–86, use the given derivative $f'$ to find any local extrema and inflection points of f and sketch a possible graph without first finding a formula for f.
$f' (x) = x^{3} - 3 x^{2} + 3 x$