Question

Observe the following statement.

Assertion (A): $f\left(x\right)=2x^3-9x^2+12x-3$ is increasing outside the interval $\left(1,2\right)$,

Reason (R): $f\left(x\right)<0$ for $x\in \left(1,2\right)$.

Then, which of the following is correct?

• A. Both A and R are correct and R is the correct reason for A
• B. Both A and R are correct and R is not the correct reason for A
• C. A is correct but r is incorrect
• D. A is incorrect but R is correct

Answer 1

The correct option is B

Both A and R are correct and R is not the correct reason for A

Explanation for the correct option(s)

Find the which of the following statement is correct.

Consider given data as,

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

For increasing $f\text{'}\left(x\right)>0$

Then,

$$\begin{gathered} f\text{'}\left(x\right)=6x^2-18x+12x>0 \\ f\text{'}\left(x\right)=6\left(x^2-3x+2x\right)>0 \\ f\text{'}\left(x\right)=6\left(x^2-2x-1x+2x\right)>0 \\ f\text{'}\left(x\right)=6\left(x\left(x-2\right)-1\left(x-2\right)\right)>0 \\ f\text{'}\left(x\right)=\left(x-1\right)\left(x-2\right)>0 \end{gathered}$$

For, $x\in \left(-\infty ,a\right)\cup \left(b,\infty \right)$

So, assertion is true

$$\begin{gathered} \left(x-a\right)\left(x-b\right)<0 \\ x\in \left(a,b\right) \end{gathered}$$

For decreasing

$$\begin{gathered} f\text{'}\left(x\right)<0 \\ \left(x-1\right)\left(x-2\right)<0 \\ x\in \left(1,2\right) \end{gathered}$$

So, Reason is true.

Hence, option B is correct.