Calculus (3rd Edition)

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

Chapter 2 - Limits - Chapter Review Exercises - Page 96: 71


The curves $y=x^{2}$ and $y=\cos x$ intersect.

Work Step by Step

Let $f(x)=x^{2}-\cos x .$ Note that any root of $f(x)$ corresponds to a point of intersection between the curves $y=x^{2}$ and $y=\cos x .$ Now, $f(x)$ is continuous over the interval $$\left[0, \frac{\pi}{2}\right], f(0)=-1\lt0$$ and $$f\left(\frac{\pi}{2}\right)=\frac{\pi^{2}}{4}\gt0$$ Therefore, by the Intermediate Value Theorem, there exists a $c \in\left(0, \frac{\pi}{2}\right)$ such that $f(c)=0 ;$ consequently, the curves $y=x^{2}$ and $y=\cos x$ intersect.
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