Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 4: Applications of Derivatives - Section 4.2 - The Mean Value Theorem - Exercises 4.2 - Page 198: 16

Answer

Proof given below.

Work Step by Step

$f''$ is contiuous $\Rightarrow f'$ is differentiable on $(a,b) $(and continuous on $[a,b].$) $\Rightarrow f$ is differentiable on $(a,b) $(and continuous on $[a,b].$) There exist $a \leq c_{1} \ \lt c_{2} \ \lt c_{3}\leq b$ such that $f(c_{1})=f(c_{2})=f(c_{3})=0$. f satisfies conditions of Rolle's theorem on $\ [c_{1},c_{2}]\ \Rightarrow$ there is a $d_{1}\in(c_{1},c_{2})$ such that $f'(d_{1})=0$ f satisfies conditions of Rolle's theorem on $\ [c_{2},c_{3}]\ \Rightarrow$ there is a $d_{2}\in(c_{2},c_{3})$ such that $f'(d_{2})=0$ $f'$ satisfies conditions of Rolle's theorem on $\ [d_{1},d_{2}]\ \Rightarrow$ there is an $e\in(d_{1},d_{2})$ such that $f''(e)=0$, which proves the statement. --- To generalize, if $f^{(n)}$ is continuous on $[a,b]$ and f has $n+1$ zeros in the interval, then $f^{(n)}$ has at least one zero in $(a,b).$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.