#55.

Suppose f is a twice-differentiable function on the set of real numbers and that f(0), f'(0), and f''(0) are all negative. Suppose f''has three of the following properties.

I. It is increasing on the interval [0, inf).

II. It has a unique zero in the interval [0, inf).

III. It Is unbounded on the interval [0, inf).

Which of the same properties does f necessarily have?

(A) I

(B) II

(C) III

(D) II and III (answer)

(E) I, II and III

#57.

Let R be the field of real numbers and R [x] the ring of polynomials in x with coefficients in R. Which of the following subsets of R [x] is a subring of R [x]?

I. All polynomials whose coefficient of x is zero

II. All polynomials whose degree is an even integer, together with zero polynomial

III. All polynomials whose coefficients are rational numbers

(A) I

(B) II

(C) I and III (answer)

(D) II and III

(E) I, II and III