Question

Suppose f is continuous on [a,b]. Then there is a point c in (a,b) such that f(c) = [f(a)+f(b)]/2. Choose the correct answer below.

A. The statement is true; since c is in (a,b) then it is always true that [f(a)+f(b)]/2 is in (a,b).
B. The statement is false; if f(a)=f(b) and the function value at every point in the interval is either less than or greater than the function value at the endpoints, then f(c)=f(a)=f(b), which would not exist in the interval (a,b).
C. The statement is true by the Intermediate Value Theorem.
D. The statement is false; f(c) can never equal [f(a)+f(b)]/2 in any function.

Answer 1

The statement is true by the Intermediate Value Theorem.

Step-by-step explanation:

The correct answer is C. The statement is true by the Intermediate Value Theorem.

According to the Intermediate Value Theorem, if a function f is continuous on a closed interval [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k.

In this case, the number [f(a) + f(b)]/2 is between f(a) and f(b) because it is the average of the function values at the endpoints of the interval. Therefore, there exists at least one point c in (a, b) such that f(c) = [f(a) + f(b)]/2.