Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. $f ( x ) = 2 \sqrt { x }$ , $[ 0,9 ]$
A) $c =$ $\frac { 9 } { 4 }$
B) $c = 5$
C) $c = 0$
D) $c = \frac { 1 } { 3 }$
E) None of these

Question

Quizplus · Accepted Answer

The Mean Value Theorem states that there exists a number $c$ in the open interval $(0, 9)$ such that $f'(c) = \frac{f(9) - f(0)}{9 - 0}$. For $f(x) = 2\sqrt{x}$, $f'(x) = \frac{1}{\sqrt{x}}$. Solving $\frac{1}{\sqrt{c}} = \frac{2\sqrt{9} - 2\sqrt{0}}{9}$ gives $c = \frac{9}{4}$.

Find the Number C That Satisfies the Conclusion of the Mean