Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 5 - Graphs and the Derivative - 5.3 Higher Derivatives, Concavity, and the Second Derivative Test - 5.3 Exercises - Page 284: 41


$$ f(x)=18x-18e^{-x} $$ $f(x)$ is never concave upward $f(x)$ always concave downward No Inflection points.

Work Step by Step

$$ f(x)=18x-18e^{-x} $$ The first derivative is $$ \begin{aligned} f^{\prime}(x) &=18-(-1)18e^{-x}\\ &=18+18e^{-x} , \end{aligned} $$ and the second derivative is $$ \begin{aligned} f^{\prime\prime}(x) &=0+18(-1)e^{-x} ,\\ &=-18e^{-x} \end{aligned} $$ Since $$ \begin{aligned} f^{\prime\prime}(x) &=-18e^{-x} \lt 0 \end{aligned} $$ for all $x$, so function $f$ is always concave downward and never concave upward and also there are no points of inflection .
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