Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).

Let $f (x) = x^3- 12x^2 + 36x + 17$.

Assertion (A): $f$ is strictly increasing in $(-∞, 2] ∪ [6, ∞)$
Reason (R): $f$ is strictly decreasing in [2, 6].

Select the correct answer from the options given below:

Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

The correct answer is Option (2) → Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

$f(x) = x^3- 12x^2 + 36x + 17$

$⇒f'(x) = 3x^2-24x+36=3(x^2-8x+12)$

$=3(x-2) (x −6)$

For function to be strictly increasing $f'(x) > 0$ i.e. $3(x-2) (x-6) > 0$

$⇒x ∈ (-∞, 2) ∪ (6, ∞)$

⇒ f is strictly increasing in $(-∞, 2) ∪ (6, ∞)$

∴ Assertion is true.

For function to be strictly decreasing $f'(x) <0$

i.e. $3(x – 2) (x – 6)<0=x∈  (2,6)$

⇒ f is strictly decreasing in [2, 6]

∴ Reason is true.

Hence, Assertion and Reason both are true but Reason is not the correct explanation of Assertion.