First slide

Theorems of probability

Question

$Consider f (x) = x^{3} + a x^{2} + b x + c . Parameters a, b, c are chosen, respectively, by throwing a die three times. Then$
$the probability that f (x) is an increasing function is$

Moderate

A

$\frac{5}{36}$
B

$\frac{8}{36}$
C

$\frac{4}{9}$
D

$\frac{1}{3}$

Solution

$f^{'} (x) = 3 x^{2} + 2 a x + 9$
$y = f (x) is increasing$
$\Rightarrow f^{'} (x) \geq 0, \forall x and for f^{'} (x) = 0 should not form an interval$
$\Rightarrow (2 a)^{2} - 4 \times 3 \times b \leq 0 \Rightarrow a^{2} - 3 b \leq 0$
This 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, the
required probability is 16/36 = 4/9

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