Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 3 - Applications of Differentiation - 3.9 Antiderivatives - 3.9 Exercises - Page 283: 25

Answer

$$ f(x) =2 x^{2}-\frac{9}{28} x^{7 / 3}+C x+D $$ where $C,D$ are arbitrary constants .

Work Step by Step

$$ f^{\prime \prime}(x)=4-\sqrt[3] {x} $$ The general anti-derivative of $ f^{\prime \prime}(x)=4-\sqrt[3] {x} $ is $$ f^{\prime}(x) =4x-\frac{3}{4} x^{\frac{4}{3}}+C $$ Using the anti-differentiation rules once more, we find that $$ \begin{split} f(x) &=4 \cdot \frac{1}{2} x^{2}-\frac{3}{4} \cdot \frac{3}{7} x^{7 / 3}+C x+D \\ &=2 x^{2}-\frac{9}{28} x^{7 / 3}+C x+D \end{split} $$ where $C,D$ are arbitrary constants .
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