A

11. Let f be a differentiable function on the open interval (a,b). Which of the following statements must be true?

I. f is continuous on the closed interval [a,b].

II. f is bounded on the open interval (a,b).

III. If a < a₁ < b₁ < b, and f(a₁) < 0 < f(b₁), then there

is a number c such that a₁ < c < b₁ and f(c) = 0

A. I and II only	B. I and III only	C. II and III only
D. I, II, and III	E. The correct answer is not given by A, B, C, or D.

Solution: The answer is E

I and II are false. The function f(x) = 1/x, 0 < x < 1, is a conterexample.

Statement III is true. Apply the Intermediate Value Theorem to f on the closed interval [a₁, b₁].