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