Question

Function f(x) = 2x³ − 9x² + 12x + 29 is monotonically decreasing when
(a) x < 2
(b) x > 2
(c) x > 3
(d) 1 < x < 2

Answer 1

(d) 1 < x < 2

$$\begin{gathered} f\left(x\right)=2x^3-9x^2+12x+29 \\ f\text{'}\left(x\right)=6x^2-18x+12 \\ =6\left(x^2-3x+2\right) \\ =6\left(x-1\right)\left(x-2\right) \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be decreasing, we must have} \\ f\text{'}\left(x\right)<0 \\ \Rightarrow 6\left(x-1\right)\left(x-2\right)<0 \\ \Rightarrow \left(x-1\right)\left(x-2\right)<0\left[\mathrm{Since}6>0,6\left(x-1\right)\left(x-2\right)<0\Rightarrow \left(x-1\right)\left(x-2\right)<0\right] \\ \Rightarrow 1<x<2 \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is decreasing for}1<x<2. \end{gathered}$$