Increasing and Decreasing Functions — MATHS STD 12 Science — Question

$f(x) = 2x^3 - 9x^2 + 12x + 29$
$\Rightarrow f'(x) = 6x^2 - 18x + 12$
$\Rightarrow f'(x) = 6(x^2 - 3x + 2)$
$\Rightarrow f'(x) = 6(x - 1)(x - 2)$
For $f(x)$ to be decreasing, we must have
$ f'(x) < 0$
$\Rightarrow 6(x - 1)(x - 2) < 0$
$\Rightarrow (x - 1)(x - 2) < 0$
$[$Since, $6 > 0, 6(x - 1)(x - 2) < 0 \Rightarrow (x - 1)(x - 2) < 0]$
$\Rightarrow 1 < x < 2$
So, $f(x)$ is decreasing for $1 < x < 2.$