Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.8 Intermediate Value Theorem - Exercises - Page 86: 15


$f(0.1)f(1)<0,f$ continuous$\Rightarrow \exists x_0\in(0.1,1),f(x_0)=0$

Work Step by Step

We are given the function: $f(x)=e^x+\ln x$ First we determine the domain of $f(x)$. The domain of $e^x$ is $(-\infty,\infty)$, while the domain of $\ln x$ is $(0,\infty)$,therefore the domain of $f$ is: $(-\infty,\infty)\cap (0,\infty)=(0,\infty)$ Choose two points in the domain and compute the value of the function in them: $f(0.1)=e^{0.1}+\ln 0.1\approx -1.197$ $f(1)=e^{1}+\ln 1\approx 2.718$ As $f(x)$ is continuous on $(0.1,1)$ and $f(0.1)$ and $f(1)$ have opposite signs, it means that the function has a zero between 0.1 and 1. We can tell that there is a zero between $0.1$ and $1$.
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