Question

Let A denotes the area bounded by $y=(lnx{)}^2$ and lines y = 0 and $x=e^2$ then -

• A. $A<4e^2-3e-1$
• B. $A<4(e^2-1)$
• C. $A>5(e^2-1)$
• D. $A>\frac{5}{2}e(e-1)$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $A<4e^2-3e-1$
B $A<4(e^2-1)$
D $A>\frac{5}{2}e(e-1)$

$y=(lnx{)}^2$
$\frac{dy}{dx}=\frac{2lnx}{x},y^{{}^{\prime \prime }}=2\left[\frac{1-lnx}{x^2}\right]=0$
$x=e\to$ Point of inflection

So $\text{Area}<Ar\left(ABFG\right)+Ar\left(BCDE\right)$
$A<(e-1)\times 1+(e^2-e)\times 4$
$A<4e^2-3e-1$ (Option A)
$A<Ar\left(ACDJ\right)$
$A<(e^2-1)\times 4$
$A<4e^2-4$
$A>Ar\left(\text{trapezium}BCDF\right)$
$A>\frac{1}{2}(1+4)(e^2-e)$
$A>\frac{5}{2}e(e-1)$