University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 4 - Section 4.1 - Extreme Values of Functions - Exercises - Page 216: 58

Answer

Function $y$ has the absolute and local minimum value of $-4$ at $x=4$ and the local maximum value of $0$ at $x=0$. There is no absolute maximum value.
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Work Step by Step

$$y=x-4\sqrt x=x-4x^{1/2}$$ - Domain: $[0,\infty)$ 1) Find all the critical points: Find the derivative of the function: $$y'=1-4\times\frac{1}{2}x^{-1/2}=1-2x^{-1/2}$$ $$y'=1-\frac{2}{\sqrt x}$$ - For $y'=0$: $$1-\frac{2}{\sqrt x}=0$$ $$\frac{2}{\sqrt x}=1$$ $$\sqrt x=2$$ $$x=4$$ - For $y'$ to be undefined, $x=0$. So $x=0$ and $x=4$ are the critical points of function $y$. 2) Evaluate function $y$ at the critical points: - For $x=0$: $$y=0-4\sqrt0=0$$ - For $x=4$: $$y=4-4\sqrt4=4-4\times2=4-8=-4$$ Now check the graph of function $y=x-4\sqrt x$. We see that function $y$ has the absolute and local minimum value of $-4$ at $x=4$ and the local maximum value of $0$ at $x=0$. There is no absolute maximum value since as $x\to\infty$, $y\to\infty$ as well.
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