Question

Given a function f(x), which of the following statements must be shown to be true before we can apply the conclusion of Rolle's Theorem? One or more of the answer options may be correct.

A. The function f is continuous on the closed interval [a,b].
B. The function f is differentiable on the open intervat (a,b). f(a)=f(b)
C. There is a number c in (a,b) such that f′(c)=0.
D. There is a number c in (a,b) such that f '(c)=(f(b)-f(a))/(b-a)

Answer 1

To apply Rolle's Theorem, the function must be continuous on the closed interval [a,b] and differentiable on the open interval (a,b), with f(a) equaling f(b). Conditions C and D are results from applying Rolle's Theorem, not preconditions for its application.

Step-by-step explanation:

The student has asked which conditions must be met before we can apply the conclusion of Rolle's Theorem. The theorem states that if certain conditions are met, there is at least one number c in the open interval (a, b) such that the derivative of the function f at c is equal to zero, f'(c) = 0. Before applying Rolle's Theorem, the following must be true:

• A. The function f is continuous on the closed interval [a,b].
• B. The function f is differentiable on the open interval (a,b).
• f(a) equals to f(b) which is not explicitly listed as one of the options, but it is usually a part of the general statement of the theorem. The other conditions listed in the question C and D are consequences of the theorem, not preconditions.

Rolle's Theorem is widely used in various mathematical applications, such as proving the mean value theorem, which itself is a cornerstone in analysis and calculus.