Find the interval(s) on which the function $f (x) = 2 x^{3} - 9 x^{2} + 12 x - 5$ is increasing.

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

Answer

(d) Both

(- \infty, 1)

and

(2, \infty)

The correct answer is (d) Both $(- \infty, 1)$ and $(2, \infty)$ — Hence $f (x)$ is increasing on $(- \infty, 1)$ and $(2, \infty)$ .

Step-by-step solution

Compute the derivative: $f^{'} (x) = 6 x^{2} - 18 x + 12$ .
Factor the derivative: $f^{'} (x) = 6 (x - 1) (x - 2)$ .
Determine the sign of $f^{'} (x)$ in the intervals defined by the critical points $x = 1$ and $x = 2$ .
For $x < 1$ , both factors are negative, their product is positive, so $f^{'} (x) > 0$ .
For $1 < x < 2$ , the factors have opposite signs, so $f^{'} (x) < 0$ .
For $x > 2$ , both factors are positive, so $f^{'} (x) > 0$ .
Hence $f (x)$ is increasing on $(- \infty, 1)$ and $(2, \infty)$ .

Common mistake: Missing one of the intervals by forgetting that the derivative is positive also for

x < 1

Concept tested: Increasing intervals of a polynomial