Consider f(x)=x3+ax2+bx+c. Parameters a,b,c are  chosen, respectively, by throwing a die three times. Then  the probability that f(x) is an increasing function is

f′(x)=3x2+2ax+9y=f(x) is increasing ⇒ f′(x)≥0,∀x and for f′(x)=0 should not form an interval ⇒ (2a)2−4×3×b≤0⇒a2−3b≤0This is true for exactly 16 ordered pairs (a, b), 1 a a, b < 6,namely (1, 1), (1,2), (1, 3), (1,4), (1,5), (1, 6), (2,2), (2,3),(2,4), (2,5), (2,6), (3, 3), (3,4), (3, 6), and (4,6). Thus, therequired probability is 16/36 = 4/9