Statement-1: Let f ( x )= x 3- ax 2+ bx +6 where a , b ∈ 1,2,3,4,5,6 . The probability that f(x) is strictly increasing function is (4/9). Statement-2: If y=g(x) is differentiable function in (-∞, ∞) then g(x) is strictly increasing provided g prime(x) ≥ 0 where sign of equality holds at discrete point(s).

Question

Statement-1: Let f ( x )= x 3- ax 2+ bx +6 where a , b ∈ 1,2,3,4,5,6 . The probability that f(x) is strictly increasing function is (4/9). Statement-2: If y=g(x) is differentiable function in (-∞, ∞) then g(x) is strictly increasing provided g prime(x) ≥ 0 where sign of equality holds at discrete point(s). (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true

Tardigrade Admin · Accepted Answer

We have f prime(x)=3 x2-2 a x+b Now y=f(x) is increasing ⇒ f prime(x) ≥ 0 ∀ x and for f prime(x)=0 should not form an interval. ⇒(2 a )2-4 × 3 × b ≤ 0 ⇒ a 2-3 b ≤ 0 This is possible 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) &(4,6) Thus, required probability =(16/36)=(4/9)