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?

1 Answer
Jun 17, 2015

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 existsat least one c in (0,9) : f'(c)=0
(Notice that Rolle's theorem doesn't give you the exact number of cs nor their value)

So we have to find out that cs, 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]}