Which of the following intervals is the function $f (x) = 2 x^{3} - 9 x^{2} + 12 x - 4$ increasing?

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

Answer

(d)

(3, \infty)

The correct answer is (d) $(3, \infty)$ — Select option (d).

Step-by-step solution

Find the derivative: $f^{'} (x) = 6 x^{2} - 18 x + 12$ .
Factor the derivative: $f^{'} (x) = 6 (x^{2} - 3 x + 2) = 6 (x - 1) (x - 2)$ .
The sign of $f^{'} (x)$ changes at $x = 1$ and $x = 2$ . Create sign chart:
- For $x < 1$ , both factors are negative, product positive, so $f^{'} (x) > 0$ .
- For $1 < x < 2$ , one factor positive, one negative, product negative, so $f^{'} (x) < 0$ .
- For $x > 2$ , both factors positive, product positive, so $f^{'} (x) > 0$ .
Hence $f$ is increasing on $(- \infty, 1)$ and $(2, \infty)$ . Among the given options only $(3, \infty)$ lies in an increasing interval.
Select option (d).

Common mistake: Incorrectly factoring the derivative or missing one of the critical points

Concept tested: Increasing intervals of a polynomial