Observe the following statements A : $f(x)=2 x^3-9 x^2+12 x-3$ is increasing outside the interval $(1,2)$ $\mathrm{R}: f^{\prime}(x)

Question

Observe the following statements A : $f(x)=2 x^3-9 x^2+12 x-3$ is increasing outside the interval $(1,2)$ $\mathrm{R}: f^{\prime}(x) < 0$ for $x \in(1,2)$. Then, which of the following is true?

Aicharya · Accepted Answer

Statement A

For increasing function, $f^{\prime}(x)>0$ $\begin{array}{lcrl} & \therefore & & 6\left(x^2-3 x+2\right)>0 \\ \Rightarrow & & 6(x-2)(x-1)>0 \\ \Rightarrow & & x < 1 \text { and } x>2 \end{array}$ $\therefore f(x)$ is increasing outside the interval $(1,2)$, therefore it is true statement. From Eq. (ii) $f^{\prime}(x)=6 x^2-18 x+12$ for decreasing $\begin{array}{lcc} & & f^{\prime}(x) < 0 \\ \Rightarrow & & 6(x-2)(x-1) < 0 \\ \Rightarrow & x>1 \text { and } x < 2 \\ \therefore & f(x) \text { is decreasing in }(1,2) . \end{array}$ $\therefore \quad A$ and $R$ are both true, but $R$ is not the correct reason.$

Observe the following statements A : $f(x)=2 x^3-9 x^2+12 x-3$ is increasing outside the interval $(1,2)$…

Solution