Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 3 - The Derivative In Graphing And Applications - 3.2 Analysis Of Functions II: Relative Extrema; Graphing Polynomials - Exercises Set 3.2 - Page 205: 12

Answer

Stationary: $x=\frac{3}{4}$ and $x=0$ Critical: $x=1$ and $x=0$ and $x=\frac{3}{4}$

Work Step by Step

First derivative \[ \begin{array}{l} 2 x(-1+x)^{2 / 3}+\frac{2}{3} x^{2}(-1+x)^{-1 / 3}= f^{\prime}(x) \\ =\frac{2 x^{2}+6 x(-1+x)}{3 \sqrt[3]{-1+x}} \end{array} \] Points where the derivative is zero are stationary points. \[ \frac{-6 x +8 x^{2}}{3 \sqrt[3]{-1+x}}=0 \] A fraction is zero if the numerator is zero \[ (-3+4 x)2 x=0 \] Roots \[ x= \frac{3}{4} \text { and } x=0 \] Points where the tangent line is horizontal (thus, the derivative is zero) are critical points. \[ x=\frac{3}{4} \text { and } x=1 \text { and } x=0 \] Points where the first derivative is zero are stationary points. \[ x=\frac{3}{4} \text { and } x=0 \]
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