Calculus 8th Edition

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

Chapter 3 - Applications of Differentiation - 3.3 How Derivatives Affect the Shape of a Graph - 3.3 Exercises - Page 228: 27

Answer

See graph

Work Step by Step

$f(0)$ = $f'(0)$ = $0\Rightarrow$ the graph of $f$ passes through the origin and has a horizontal tangent there $f'(2)$ = $f'(4)$ = $f'(6)$ = $0\Rightarrow$ horizontal tangents at $x$ = $2,4,6$ $f'(x)$ $\gt$ $0$ if $0$ $\lt$ $x$ $\lt$ $2$ or $4$ $\lt$ $x$ $\lt$ $6\Rightarrow f$ increasing on $(0,2)\cup(4,6)$ $f'(x)$ $\lt$ $0$ if $2$ $\lt$ $x$ $\lt$ $4$ or $x$ $\gt$ $6\Rightarrow f$ decreasing on $(2,4)\cup(6,\infty)$ $f''(x)$ $\gt$ $0$ if $0$ $\lt$ $x$ $\lt$ $1$ or $3$ $\lt$ $x$ $\lt$ $5\Rightarrow f$ is concave upward on $(0,1)\cup(3,5)$ $f''(x)$ $\lt$ $0$ if $1$ $\lt$ $x$ $\lt$ $3$ or $x$ $\gt$ $5\Rightarrow f$ is concave downward on $(1,3)\cup(5,\infty)$ $f(-x)$ = $f(x)\Rightarrow f$ is even and the graph is symmetric about the $y$-axis
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