Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 4 - Section 4.6 - Graphing with Calculus and Calculators - 4.6 Exercises - Page 329: 1

Answer

See attached graphs. Please see below for explanations. .

Work Step by Step

(graphed with desmos.com) Row 1, graph 1: $f(x)$, zoomed out sufficiently to roughly see all areas "of interest". Row 1, graph $2$: We graph the derivative; note the zeros of $f'(x)$ and the intervals in which $f'(x)$ is positive or negative. From these data, we determine where $f(x)$ rises/falls, and where the local extremes lie. $\left[\begin{array}{cccccccccc} & -\infty & & -1.5 & & 0.0358 & & 2.841 & & 2.623 & & \infty\\ f'(x) & & + & 0 & - & 0 & + & 0 & - & 0 & + & \\ f(x) & & \nearrow & max & \searrow & min & \nearrow & max & \searrow & min & \nearrow & \end{array}\right]$ Row 1, graph 3: combined graphs of $f(x)$ and $f'(x)$, with points of extrema. Row 2: graphs of $f''(x)$ and $f(x)$, with vertical lines where $f''$ is zero. This gives us inflection points at $x=-0.888, 1.153,$ and $2.735$. $\left[\begin{array}{ccccccccc} & -\infty & & -0.888 & & 1.153 & & 2.735 & & \infty\\ f''(x) & & - & 0 & + & 0 & - & 0 & + & \\ f(x) & & \cap & infl & \cup & infl & \cap & infl & \cup & \end{array}\right]$ The graph in row 2 also gives points of inflection. Row 3: Zoom-in around the local minimum at $x=0.0358$ and around the inflection point at $x=2.735.$
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