1. Tardigrade
  2. Question
  3. Mathematics
  4. 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).

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Q. Statement-1: Let $f ( x )= x ^3- ax ^2+ bx +6$ where $a , b \in\{1,2,3,4,5,6\}$. The probability that $f(x)$ is strictly increasing function is $\frac{4}{9}$.
Statement-2: If $y=g(x)$ is differentiable function in $(-\infty, \infty)$ then $g(x)$ is strictly increasing provided $g^{\prime}(x) \geq 0$ where sign of equality holds at discrete point(s).

Probability - Part 2

Solution:

We have $f^{\prime}(x)=3 x^2-2 a x+b$
Now $y=f(x)$ is increasing $\Rightarrow f^{\prime}(x) \geq 0 \forall x$ and for $f^{\prime}(x)=0$ should not form an interval. $\Rightarrow(2 a )^2-4 \times 3 \times b \leq 0 \quad \Rightarrow a ^2-3 b \leq 0$
This is possible for exactly 16 ordered pairs $(a, b), 1 \leq a, b \leq 6$ 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 $=\frac{16}{36}=\frac{4}{9}$