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?

##c=9/4##

1) ##f## is continuous in ##[0,9]##, obvious 2) ##f## is derivable in ##(0,9)##, also obvious, its derivative is ##1/(2sqrt(x)) - 1/3##, which is well defined for all x in ##(0,9)## (NB: zero not included)

3) ##f(0)=0, f(9)=3-9/3=0##, so ##f(0)=f(9)##

So Rolle's theorem states that ##exists##at least one ##c in (0,9) : f'(c)=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 ##(0,9)## of ##f'(x)=0## i.e.

##1/(2sqrt(c))-1/3=0 => 1=2/3sqrt(c) => c=(3/2)^2=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]}