Q.50. Probability, Polynomial function

Question 50.
Suppose f(x)=x^3+ax^+bx+c where a,b,c are chosen respectively by throwing a die three times.?
Then the probability that f(x) is an increasing function, is-

Answer 50.
f(x) = x^3 + ax^2 + bx + c
=> f '(x) = 3x^2 + 2ax + b
For f(x) to be an increasing function, f '(x) ≥ 0
=> 3x^2 + 2ax + b ≥ 0
=> f '(x) has 0 or 1 real root

=> 4a^2 - 12b ≤ 0 => a^2 ≤ 3b
=> b = 1 and a = 1
b = 2 and a = 1, 2
b = 3 and a = 1, 2, 3
b = 4 and a = 1, 2, 3,
b = 5 and a = 1, 2, 3,
b = 6 and a = 1, 2, 3, 4
=> required probab ility
= (1/6) * [(1/6) + (2/6) + 3*(3/6) + (4/6)]
= 4/9.

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