The function $\mathrm{f}(x)=2 x^3-9 x^2+12 x+2$ is decreasing in

The function $\mathrm{f}(x)=2 x^3-9 x^2+12 x+2$ is decreasing in
  1. $1 \lt x \lt 2$
  2. $x \lt 1$ or $x \gt 2$
  3. $x \lt -1$ or $x \gt -2$
  4. $-2 \lt x \lt -1$

Solution

$\begin{aligned} & \mathrm{f}(x)=2 x^3-9 x^2+12 x+2 \\ \therefore \quad & \mathrm{f}^{\prime}(x)=6 x^2-18 x+12 \end{aligned}$
For decreasing function, $\mathrm{f}^{\prime}(x) \lt 0$ $\begin{aligned} & \Rightarrow 6 x^2-18 x+12 \lt 0 \\ & \Rightarrow x^2-3 x+2 \lt 0 \\ & \Rightarrow(x-1)(x-2) \lt 0 \\ & \Rightarrow x \in(1,2) \end{aligned}$

Asked in: MHT CET 2024 (15 May Shift 2)

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