Question

Assertion :Statement-1: Let $f\left(x\right)=\frac{20}{4x^3-9x^2+6x}$ then the function $f$ is unbounded. Reason: Statement-2 : $f$ increases on $(1/2,1)$ and decreases on $(1,\infty )\cup (-\infty ,1/2).$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$f\left(x\right)=\frac{20}{4x^3-9x^2+6x}$
let $g\left(x\right)=4x^3-9x^2+6x$
$\Rightarrow g^{\prime }\left(x\right)=6(2x^2-3x+1)=6(2x-1)(x-1)=0$
$f\left(x\right)$ increases when $g^{\prime }\left(x\right)<0$
$\Rightarrow x\in (\frac{1}{2},1)$
$f\left(x\right)$ decreases when $g^{\prime }\left(x\right)>0$
$\Rightarrow x\in (1,\infty )\cup (-\infty ,1/2)$
Since, $f\left(x\right)$ increases in $(1/2,1)$
Therefore, min$f\left(x\right)=f\left(\frac{1}{2}\right)=16$ and max$f\left(x\right)=f\left(2\right)=20$