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+by = 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, 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, 5), (3, 6), and (4,6). Thus, the required probability is 16/36 = 4/9