Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 5 - Graphs and the Derivative - 5.3 Higher Derivatives, Concavity, and the Second Derivative Test - 5.3 Exercises: 4

Answer

\[\,\,\,\,\,\, - 1,35\]

Work Step by Step

\[\begin{gathered} f\,\left( x \right) = - {x^4} + 7{x^3} - \frac{{{x^2}}}{2} \hfill \\ Find\,\,the\,\,derivative\,\,of\,\,the\,\,function \hfill \\ {f^,}\,\left( x \right) = \,\,{\left[ { - {x^4} + 7{x^3} - \frac{{{x^2}}}{2}} \right]^,} \hfill \\ Use\,\,the\,\,power\,\,rule \hfill \\ {f^,}\,\left( x \right) = - 4{x^3} + 21{x^2} - x \hfill \\ Find\,\,the\,\,\sec ond\,\,derivative \hfill \\ {f^{\,,,}}\,\left( x \right) = \,\left( { - 4{x^3} + 21{x^2} - x} \right) \hfill \\ {f^{\,,,}}\,\left( x \right) = - 12{x^2} + 42x - 1 \hfill \\ Evaluate\,\,{f^{,,}}\,\left( 0 \right)\,\,and\,{f^{,,}}\,\left( 2 \right)\,\, \hfill \\ {f^{\,,,}}\,\left( 0 \right) = - 12\,{\left( 0 \right)^2} + 42\,\left( 0 \right) - 1 = - 1 \hfill \\ {f^{\,,,}}\,\left( 2 \right) = - 12\,{\left( 2 \right)^2} + 42\,\left( 2 \right) - 1 = 35 \hfill \\ \end{gathered} \]
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