Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 2 - Section 2.3 - Getting Information from the Graph of a Function - 2.3 Exercises - Page 182: 69

Answer

(a) see below. (b) $g(x)=\sqrt {x^4+x^2-6x+9}$ (c) $g(1)=\sqrt 5$

Work Step by Step

(a) $g(x)=\sqrt {f(x)}, f(x)\geq0$, let f($x_0)$ be a minimum of $f(x)$ in an open interval $(a,b)$ such that for any point $x$ in $(a,b)$ except $x_0$, we have $f(x_0)\lt f(x)$. This will also be true for $g(x)$ as $g(x_0)=\sqrt {f(x_0)}\lt \sqrt {f(x)}=g(x)$. We can also show it is true for a maximum. (b) $g(x)=\sqrt {(x-3)^2+(x^2-0)^2}=\sqrt {x^4+x^2-6x+9}$ (c) Base on (a), the minimum of the function inside the radical can be found as $f(1)=5$, and the minimum of $g(1)=\sqrt 5$ can thus be obtained.
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