Question

Observe the following statements
Assertion (A) $f(x)=2 x^3-9 x^2+12 x-3$ 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)

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