rolle's theorem how do you solve f(x)= x^1/2 - 1/3x , [0,9]

Question

Using rolle's theorem how do you solve f(x)= x^1/2 - 1/3x , [0,9]

Answer 1

At the extremes of the range f(x)=0. So by Rolle’s Theorem there must be at least one value where f(x)≠0 in the interval. If we put x=1 f(1)=⅔ and midrange f(4)=2-4/3=⅔. Both values are positive so the curve does not cross the x axis because there is no change of sign. f(8.41)=2.9-8.41/3>0 and 8.41 is close to the end of the interval. Rolle’s Theorem maintains that only if there is a change of sign does the curve cross the x axis. Therefore we conclude that the solutions are x=0 and x=9, other values in the range being positive.