Which of the following intervals is the function $f (x) = 3 x^{3} - 12 x^{2} + 9 x + 5$ increasing? (a) $(- in f t y, 1)$ (b) $(1, 2)$ (c) $(2, 3)$ (d) $(3, in f t y)$

easy MCQ 1 mark Class 12 · Mathematics · Application of Derivatives

Answer

(b)

(1, 2)

The correct answer is (b) $(1, 2)$ — Select option (b).

Step-by-step solution

Find the derivative $f^{'} (x) = 9 x^{2} - 24 x + 9$ .
Factor $f^{'} (x) = 9 (x - 1) (x - 3)$ .
Identify sign of $f^{'} (x)$ : positive when $x in (1, 3)$ .
Exclude the point $x = 3$ because derivative zero there. The function increases on $(1, 2)$ and $(2, 3)$ . The only option matching is $(1, 2)$ .
Select option (b).

Common mistake: Not simplifying the derivative and missing sign changes at critical points

Concept tested: Increasing/decreasing intervals