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  4. The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x -axis and vertices C and D lie on the parabola, y = x 2-1 below the x -axis, is

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Q. The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$ -axis and vertices $C$ and $D$ lie on the parabola, $y = x ^{2}-1$ below the $x$ -axis, is

JEE MainJEE Main 2020Application of Derivatives

Solution:

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Area $( A )=2 t \cdot\left(1- t ^{2}\right)$
$(0< t <1)$
$A =2 t -2 t ^{3}$
$\frac{ d A }{ dt }=2-6 t ^{2}$
$t =\frac{1}{\sqrt{3}} $
$\Rightarrow A _{\max }=\frac{2}{\sqrt{3}}\left(1-\frac{1}{3}\right)=\frac{4}{3 \sqrt{3}}$