Andrew D. answered • 06/18/21
Degree in applied mathematics with calculus tutoring experience
This is a little bit of a trick question. So, we have to think about what the intermediate value theorem says: that given a continuous function defined on some interval [a, b], there must be a c in (a, b) such that f(c) is in between f(a) and f(b). The key here is that c should be in the non-inclusive interval; that is, it doesn't include the endpoints. When you look at the interval created by [f(2), f(4)] it becomes trivially true that the equation has at least one solution, not actually needing it to be proved by the IVT.
So, this comes down to understanding the purpose of the IVT versus what you can glean from attempting this proof using it. If you went through the steps of using the IVT, it would become apparent that the equation had a solution. However, it would not be because of the implications of the IVT - that is, that the function will always have some intermediate value in the interval [a,b] such that f(c) is in between f(a) and f(b). It would simply be due to trivial observation.
