ahh i think I know what you mean. But it was given that x1 = k, and x2 = k+1/k^2, so x2> x1The change of sign alone doesn’t tell you which way around the inequality is. You need to know the behaviour of the function in that domain to be able make that conclusion. You are slightly saved by the fact that x2 can be explictly shown to be greater than x1 in the particular example.
i agree but i still think f(x) is continuous is important because if it isn't continuous, there doesn't have to be a root between x2 and x1You don't need to say that the function is continuous and increasing in the given domain. The question stated that K was a positive integer with x1. Since x2 is K+(1/K^2), the x2 must be greater than x1 since (1/K^2)>0. From this, it can be deduced that the root alpha must lie between x2 and x1 (with given proof of course).