Calculus: Early Transcendentals 8th Edition

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

Chapter 2 - Section 2.5 - Continuity - 2.5 Exercises - Page 125: 51


There is a real number $c$ such that $~f(c) = 1000$

Work Step by Step

Let $g(x) = x^2$ Let $h(x) = 10~sin~x$ These two functions are continuous on the interval $(-\infty, \infty)$ $f(x) = x^2+10~sin~x~~$ is also continuous on the interval $(-\infty, \infty)$, since it is the sum of two continuous functions. Let $x = 0$: $f(0) = (0)^2+10~sin~(0) = 0$ Let $x = 100~pi$: $f(100~\pi) = (100~\pi)^2+10~sin~(100~\pi) = 10,000~\pi^2$ Note that: $0 \lt 1000 \lt 10,000~\pi^2$ Since $f(x)$ is a continuous function, there must be a real number $c$ where $0 \lt c \lt 100~\pi~~$ such that $~f(c) = 1000$
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