Question:
The area (in sq. units) of the largest rectangle $\mathrm{ABCD}$ whose vertices $\mathrm{A}$ and $\mathrm{B}$ lie on the $\mathrm{x}$-axis and vertices C and D lie on the parabola, $\mathrm{y}=\mathrm{x}^{2}-1$ below the $\mathrm{x}$-axis, is :
Correct Option: 1
Solution:
Area $(\mathrm{A})=2 \mathrm{t} \cdot\left(1-\mathrm{t}^{2}\right)$
$(0<\mathrm{t}<1)$
$A=2 t-2 t^{3}$
$\frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=2-6 \mathrm{t}^{2}$
$t=\frac{1}{\sqrt{3}}$
$\Rightarrow A_{\max }=\frac{2}{\sqrt{3}}\left(1-\frac{1}{3}\right)=\frac{4}{3 \sqrt{3}}$
