Question

Function f(x) = x³ − 27x + 5 is monotonically increasing when
(a) x < −3
(b) | x | > 3
(c) x ≤ −3
(d) | x | ≥ 3

Answer 1

(d) | x | ≥ 3

$$\begin{gathered} f\left(x\right)=x^3-27x+5 \\ f\text{'}\left(x\right)=3x^2-27 \\ =3\left(x^2-9\right) \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow 3\left(x^2-9\right)>0 \\ \Rightarrow \left(x^2-9\right)>0\left[\mathrm{Since}3>0,3\left(x^2-9\right)>0\Rightarrow \left(x^2-9\right)>0|\right] \\ \Rightarrow \left(x+3\right)\left(x-3\right)>0 \\ \Rightarrow x<-3\text{or}x>3 \\ \Rightarrow \left|x\right|>3 \end{gathered}$$