I need to prove this using the mean value theorem.

1/9 < sqrt(66) - 8 < 1/8

I assumed a = 64 and b = 66 and let a function f(x) = sqrt(x) such that x lies in the interval [a, b]

Then by the mean value theorem,

f'(x) = 1/2sqrt(x) = (sqrt (66) - sqrt (64)) / 66-64 = (sqrt(66) - 8)/2

But at this point, I'm beautifully stuck, and dont know what else to do.

Any ideas?