Could someone please explain to me how I can use the intermediate value theorem to convince myself that m where [tex] m = x^5 - 1 = x^3 [/tex] exists. I have deduced that I can work in interval [1,2] using Newtons method where [tex] x^5 - 1 - x^3 = 0 [/tex] to find the answer. I even have the answer correct to four places, but part of the question is to show precisely how the intermediate value theorem proves the number exists in the positive real numbers before you do it. My main problems are 1. I am not sure exactly how to prove that the function is continuous on the interval. (I have proved its continuous at each end.) and 2. I am not sure exactly how to prove that the value for [itex] x^3 [/itex] is equal to the value for [itex] y^5 - 1 [/itex] when [itex] x=y [/itex] using **only** IVT. I mean, what in the intermediate value theorem proves that (and/or where) [itex] x^3 [/itex] crosses (or aligns with) [itex] x^5-1 [/itex]. Because I believe I am supposed to show that. Just looking at the graphs of the equations and giving it a second thought is proving that the limit --> 0 exists on [tex] x^5 - 1 - x^3 = 0 [/tex] and is equal to f(0) hence proving continuity of the equation at x=0 what I should be doing? using IVT then just on this equation would say that one value of x^5 - 1 is equal exactly to x^3. I think I have just answered my own question.