Question

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?

Answer 1

`c=9/4`

Explanation:

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]}