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: 62


$$ f(x) \geq10 x-15$$

Work Step by Step

By using the MVT, there exists $c\in[a,b]$ such that $$ f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} $$ Applying on $[2,x]$, we get \begin{align*} \frac{f(x)-f(2)}{x-2}&=f^{\prime}(c)\\ f(x)-f(2)&=(x-2) f^{\prime}(c)\\ \end{align*} Since $f'(x) \geq10$, then \begin{align*} f(x)-f(2) &\geq 10(x-2), \\ f(x)& \geq f(2)+10(x-2)\\ &\geq10 x-15 \end{align*}
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