341Fact: It attains an extremum
Implication: f'(x) = 3x2+2ax+b = 0 has real roots
And so?: This means discriminant 4(a2-3b)>0. So the 1st quantity is positive (funnily, in all your choices, its +, so this was not needed. still...)
Fact the extremum is a minima. Also the point of minimum is negative.
Implication f"(x0)>0 with x0<0 where x0 is the minimum point
And so? f"(x0) >0 means 6x0+2a>0. or x_0> -\frac{a}{3} . You can see that if a<0, we are in serious problems as that would mean x0>0. So a>0
Fact 3x2+2ax+b = 0 has a negative root. I hope that means only one negative root
Implication Not much really to add to previous sentence.
And so? so the other roots is +ve and hence product of roots is -ve. so that b<0.
otherwise both options c and d are correct
