Question

The interval on which the function f(x) = 2x³ + 9x² + 12x - 1 is decreasing is
(a) [-1, ∞) (b) [-2, -1] (c) (∞, -2] (d) [-1, 1]

Answer 1

$f\left(x\right)=2x^3+9x^2+12x-1$

$\Rightarrow f\text{'}\left(x\right)=6x^2+18x+12$

$\Rightarrow f\text{'}\left(x\right)=6\left(x^2+3x+2\right)$

$\Rightarrow f\text{'}\left(x\right)=6\left(x+2\right)\left(x+1\right)$

For f(x) to be decreasing,

$f\text{'}\left(x\right)<0$

$\Rightarrow 6\left(x+2\right)\left(x+1\right)<0$

$\Rightarrow \left[x-\left(-2\right)\right]\left[x-\left(-1\right)\right]<0$

$\Rightarrow -2<x<-1$ [For a < b, if (x − a)(x − b) < 0 ⇒ a < x < b]

Thus, the interval on which the given function f(x) is decreasing is [−2, −1].

Hence, the correct answer is option (b).