Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 3 - Applications of Differentiation - 3.4 Exercises - Page 192: 17

Answer

$$\eqalign{ & {\text{Inflection points: }}\left( { - 2, - 8} \right){\text{ and }}\left( {0,0} \right) \cr & {\text{Concave downward}}:{\text{ }}\left( { - 2,0} \right) \cr & {\text{Concave upward}}:{\text{ }}\left( { - \infty ,2} \right){\text{ and }}\left( {0,\infty } \right) \cr} $$

Work Step by Step

$$\eqalign{ & f\left( x \right) = \frac{1}{2}{x^4} + 2{x^3} \cr & {\text{Calculate the second derivative}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{1}{2}{x^4} + 2{x^3}} \right] \cr & f'\left( x \right) = 2{x^3} + 6{x^2} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {2{x^3} + 6{x^2}} \right] \cr & f''\left( x \right) = 6{x^2} + 12x \cr & {\text{Set }}f''\left( x \right) = 0 \cr & 6{x^2} + 12x = 0 \cr & 6x\left( {x + 2} \right) = 0 \cr & 6x = 0,{\text{ }}x + 2 = 0 \cr & x = 0,{\text{ }}x = - 2 \cr & {\text{Set the intervals }}\left( { - \infty , - 2} \right),\left( { - 2,0} \right),\left( {0,\infty } \right) \cr & {\text{Making a table of values }}\left( {{\text{See examples on page 188 }}} \right) \cr} $$ \[\boxed{\begin{array}{*{20}{c}} {{\text{Interval}}}&{\left( { - \infty , - 2} \right)}&{\left( { - 2,0} \right)}&{\left( {0,\infty } \right)} \\ {{\text{Test Value}}}&{x = - 4}&{x = - 1}&{x = 2} \\ {{\text{Sign of }}f''\left( x \right)}&{48 > 0}&{ - 6 < 0}&{48 > 0} \\ {{\text{Conclusion}}}&{{\text{C}}{\text{. upward}}}&{{\text{C}}{\text{. downward}}}&{{\text{C}}{\text{. upward}}} \end{array}}\] $$\eqalign{ & {\text{The inflection points occurs at }}x = - 2{\text{ and }}x = 0 \cr & f\left( { - 2} \right) = \frac{1}{2}{\left( { - 2} \right)^4} + 2{\left( { - 2} \right)^3} = - 8,{\text{ }}\left( { - 2, - 8} \right) \cr & f\left( 0 \right) = \frac{1}{2}{\left( 0 \right)^4} + 2{\left( 0 \right)^3} = 0,{\text{ }}\left( {0,0} \right) \cr & {\text{Inflection points: }}\left( { - 2, - 8} \right){\text{ and }}\left( {0,0} \right) \cr & {\text{Concave downward}}:{\text{ }}\left( { - 2,0} \right) \cr & {\text{Concave upward}}:{\text{ }}\left( { - \infty ,2} \right){\text{ and }}\left( {0,\infty } \right) \cr} $$
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