Applications of Rolle's Theorem

When Rolle's theorem is applied, is rarely to find the places where the tangent line is horizontal. Indeed, it gives no information about their location, only that they are somewhere, just as our first theorem of this section let us know conditions under which maximum and minimum values of a function could be found, without telling us how to find them. Such theorems are called existence theorems, as they tell us that certain quantities or objects exist (in a mathematical sense). These are contrasted with things like Newton's method, which are constructions, and tell us how to find approximations or exact values. Neither constructions nor existence theorems are any good alone, but instead work together to solve practical problems.