Question

If $f (x) = 3x^4 + 4x^3 - 12x^2 + 12$, then f (x) is

• A. increasing in $(- \infty , - 2)$ and in (0, 1)
• B. increasing in ( - 2, 0) and in $(1, \infty )$
• C. decreasing in ( - 2, 0) and in (0,1)
• D. decreasing in $( - \infty , - 2)$ and in $(1, \infty )$

Answer 1

Answer: (B)

Given : $f\left(x\right) = 3x^{4} + 4x^{3} - 12 x^{2} + 12$
Differentiating with respect to x, we get
$ f ' \left(x\right) = 12x^{3}+ 12x^{2}- 24x $
For f (x) to be increasing
f '(x) > 0
$\Rightarrow \, 12x^3 + 12x^2 - 24x > 0 $
$\Rightarrow \, 12x (x^2 + x - 2) > 0 $
$\Rightarrow \, 12x (x - 1) (x + 2) > 0$
$\Rightarrow \, x (x - 1) (x + 2) > 0$
$\Rightarrow \, - 2 < x < 0$ or $x > 1$