## Calculus with Applications (10th Edition)

Published by Pearson

# Chapter 5 - Graphs and the Derivative - 5.3 Higher Derivatives, Concavity, and the Second Derivative Test - 5.3 Exercises - Page 283: 18

#### Answer

$$f'''\left( x \right) = - 48x + 42,\,\,\,\,\,\,\,\,\,\,\,\,{f^{\left( 4 \right)}}\left( x \right) = - 48$$

#### Work Step by Step

\eqalign{ & f\left( x \right) = - 2{x^4} + 7{x^3} + 4{x^2} + x \cr & {\text{find the derivative of }}f\left( x \right) \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ { - 2{x^4} + 7{x^3} + 4{x^2} + x} \right] \cr & {\text{apply the power rule }}\frac{d}{{dx}}\left[ {{x^n}} \right] = {x^{n - 1}}{\text{ to each term}} \cr & f'\left( x \right) = - 2\left( {4{x^3}} \right) + 7\left( {3{x^2}} \right) + 4\left( {2x} \right) + 1 \cr & f'\left( x \right) = - 8{x^3} + 21{x^2} + 8x + 1 \cr & \cr & {\text{find the derivative of }}f'\left( x \right) \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ { - 8{x^3} + 21{x^2} + 8x + 1} \right] \cr & {\text{use the power rule}} \cr & f''\left( x \right) = - 8\left( {3{x^2}} \right) + 21\left( {2x} \right) + 8\left( 1 \right) \cr & f''\left( x \right) = - 24{x^2} + 42x + 8 + 0 \cr & \cr & {\text{find the derivative of }}f''\left( x \right) \cr & f'''\left( x \right) = \frac{d}{{dx}}\left[ { - 24{x^2} + 42x + 8} \right] \cr & {\text{use the power rule}} \cr & f'''\left( x \right) = - 24\left( {2x} \right) + 42\left( 1 \right) + 0 \cr & f'''\left( x \right) = - 48x + 42 \cr & \cr & {\text{find the derivative of }}f'''\left( x \right) \cr & {f^{\left( 4 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ { - 48x + 42} \right] \cr & {\text{then}} \cr & {f^{\left( 4 \right)}}\left( x \right) = - 48\left( 1 \right) + 0 \cr & {f^{\left( 4 \right)}}\left( x \right) = - 48 \cr}

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