Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 3 - Differentiation - 3.2 Exercises: 9

Answer

Rolle's theorem can be applied; $c=\dfrac{3}{2}.$

Work Step by Step

Since $f(x)$ is a polynomial, it is continuous for all values of $x$ and differentiable at every value of $x$. $f(0)=f(3)=0.$ Since $f(x)$ is continuous over $[0 ,3 ]$ and differentiable over $(0, 3)$, applying Rolle's Theorem over the interval $[0, 3]$ guarantees the existence of at least one value $c$ such that $0\lt c\lt 3$ and $f'(c)=0.$ $f'(x)=-2x+3\to f'(x)=0 \to c=\dfrac{3}{2}.$
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