# How do you verify that the function f(x)= (sqrt x)- 1/3 x satisfies the three hypothesis of Rolles's Theorem on the given interval [0,9] and then find all numbers (c) that satisfy the conclusion of Rolle's Theorem?

Jun 17, 2015

$c = \frac{9}{4}$

#### Explanation:

1) $f$ is continuous in $\left[0 , 9\right]$, obvious
2) $f$ is derivable in $\left(0 , 9\right)$, also obvious, its derivative is $\frac{1}{2 \sqrt{x}} - \frac{1}{3}$, which is well defined for all x in $\left(0 , 9\right)$ (NB: zero not included)

3) $f \left(0\right) = 0 , f \left(9\right) = 3 - \frac{9}{3} = 0$, so $f \left(0\right) = f \left(9\right)$

So Rolle's theorem states that $\exists$at least one $c \in \left(0 , 9\right) : f ' \left(c\right) = 0$
(Notice that Rolle's theorem doesn't give you the exact number of $c$s nor their value)

So we have to find out that $c$s, which are all the solutions in $\left(0 , 9\right)$ of $f ' \left(x\right) = 0$ i.e.

$\frac{1}{2 \sqrt{c}} - \frac{1}{3} = 0 \implies 1 = \frac{2}{3} \sqrt{c} \implies c = {\left(\frac{3}{2}\right)}^{2} = \frac{9}{4}$

So we have only one $c$

If you look at the graph you can convince yourself the answer is correct and the meaning of Rolle's theorem
(although, a graph is not a proof)

graph{sqrt(x) -1/3x [-10, 10, -5, 5]}