Consider with

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider

F_{a,b}(x)=a x^{3}+b x^{2}+c x+d

with

a\neq 0

. The graph of

F_{a,b}

is required to be concave down on

(-\infty,0)

and concave up on

(0,\infty)

. Which of the following conditions on

a

and

b

is both necessary and sufficient? (a)

a>0\text{ and }b<0

(b)

a<0\text{ and }b=0

(c)

a>0\text{ and }b=0

(d)

a<0\text{ and }b>0