Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 257: 35

Answer

$$\eqalign{ & {\text{Increasing on }}\left( { - \infty , - \frac{1}{2}} \right),\left( {0,\frac{1}{2}} \right) \cr & {\text{Decreasing on }}\left( { - \frac{1}{2},0} \right),\,\left( {\frac{1}{2},\infty } \right) \cr} $$

Work Step by Step

$$\eqalign{ & f\left( x \right) = - 2{x^4} + {x^2} + 10 \cr & {\text{Derivative}} \cr & f'\left( x \right) = - 8{x^3} + 2x \cr & {\text{Set the derivative to 0}} \cr & - 8{x^3} + 2x = 0 \cr & 2x\left( { - 4{x^2} + 1} \right) = 0 \cr & 2x\left( {1 + 2x} \right)\left( {1 - 2x} \right) = 0 \cr & {\text{Solving the equation we obtain}} \cr & x = 0,\,\,\,x = - \frac{1}{2}{\text{ and }}x = \frac{1}{2} \cr & {\text{From the critical values and the domain }}\left( { - \infty ,\infty } \right){\text{ we have}} \cr & \left( { - \infty , - \frac{1}{2}} \right),\,\,\left( { - \frac{1}{2},0} \right),\,\,\left( {0,\frac{1}{2}} \right),\,\,\left( {\frac{1}{2},\infty } \right) \cr & {\text{Now}}{\text{, we will evaluate the critical value and resume in a table}} \cr} $$ \[\begin{array}{*{20}{c}} {{\rm{Interval}}}&{{\rm{Test\ value }}\left( x \right)}&{{\rm{Sign\ of }}f'\left( x \right)}&{{\rm{Behavior \ of }}f\left( x \right)}\\ {\left( { - \infty , - \frac{1}{2}} \right)}&{ - 1}& + &{{\rm{Increasing}}}\\ {\left( { - \frac{1}{2},0} \right)}&{ - \frac{1}{4}}& - &{{\rm{Decreasing}}}\\ {\left( {0,\frac{1}{2}} \right)}&{\frac{1}{4}}& + &{{\rm{Increasing}}}\\ {\left( {\frac{1}{2},\infty } \right)}&1& - &{{\rm{Decreasing}}}\\ {}&{}&{}&{}\\ {}&{}&{}&{} \end{array}\] $$\eqalign{ & {\text{From the table we can conlude that the function is:}} \cr & {\text{Increasing on }}\left( { - \infty , - \frac{1}{2}} \right),\left( {0,\frac{1}{2}} \right) \cr & {\text{Decreasing on }}\left( { - \frac{1}{2},0} \right),\,\left( {\frac{1}{2},\infty } \right) \cr} $$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.