`f(x) = sqrt(x) - (1/3)x, [0,9]` Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers `c` that satisfy the conclusion of Rolle’s Theorem.
Rolle's Theorem requires that f is continuous on closed interval (true), f is differentiable on the open interval (also true) and f(a)=f(b).
f(0) = 0 - 0 = 0, f(9) = 3 - 9/3 = 0, also true.
Then there is c from (0, 9) such that f'(c) = 0.
`f'(c) = 1/(2sqrt(c)) - 1/3,` this is zero when
`1/(2sqrt(c)) = 1/3,` `2sqrt(c) = 3,` c = 9/4 = 2.25.
(note that f isn't differentiable at x=0 but this doesn't prevent the use of Rolle's Theorem)