Math CentralQuandaries & Queries


Question from A., a student:

When dealing with the intermediate value theorem you have the function x^2. It bounces on the axis so you can't tell if lies on the interval [a,b]. So is the ivt proven false or does the ivt not tell you all the roots for sure.


The Intermediate Value Theorem states that

if f(x) is a continuous function for a ≤ x ≤ b and f(a) < q < f(b) then there is a number p so that a < p < b and f(p) = q.

In your case since you are looking for roots you want q = 0 so the conclusion is that there is a number p so that f(p) = 0 and thus p is a root. Thus you need numbers a and b so that f(a) < 0 < f(b). But for the function f(x) = x2 there is no number a so that f(a) = a2 < 0. Thus the Intermediate Value Theorem doesn't apply to this situation.


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