Question

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

• A. Both A and R are true and R is the correct explanation for A
• B. Both A and R are true and R is not the correct explanation for A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

Answer: (B)

$f^{\prime }(x)=6x^2-18x+12$
For increasing function, $f^{\prime }(x)\ge 0$
$\therefore 6\left(x^2-3x+2\right)\ge 0$
$\Rightarrow 6(x-2)(x-1)\ge 0$
$\Rightarrow x\le 1\text{ and }x\ge 2$
$\therefore f(x)$ is increasing outside the interval $(1,2)$, therefore it is true statement.
Now, $f^{\prime }(x)<0$
$\Rightarrow 6(x-2)(x-1)<0$
$\Rightarrow 1<x<2$
$\therefore A$ and $R$ are both true but $R$ is not the correct explanation of $A$.