Calculus (3rd Edition)

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

Chapter 4 - Applications of the Derivative - 4.3 The Mean Value Theorem and Monotonicity - Exercises - Page 190: 59


$$c =\frac{a+b}{2}$$

Work Step by Step

Consider the quadratic equation $$f(x)=p x^{2}+q x+r$$ Then by using the MVT, \begin{aligned} f^{\prime}(c) &=\frac{f(b)-f(a)}{b-a} \\ &=\frac{\left(p b^{2}+q b+r\right)-\left(p a^{2}+q a+r\right)}{b-a} \\ &=\frac{p\left(b^{2}-a^{2}\right)+q(b-a)}{b-a} \\ &=\frac{p(b-a)(b+a)+q(b-a)}{b-a} \\ &=p(b+a)+q \end{aligned} On the other hand, we have $$f'(x) =2px+q $$ Then \begin{align*} f^{\prime}(c)&=p(b+a)+q=2 p c+q\\ b+a&=2 c\\ c&=\frac{a+b}{2} \end{align*}
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