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 = x² - 1 below the x – axis is________

y = x² - 1, let OB = a, B(a, 0, A(-a,0)
C = (a, a² -1),D = (-a, a² -1)
$\mathrm{CD}=\sqrt{4{\mathrm{a}}^2},\mathrm{AD}=\sqrt{0+{\left({\mathrm{a}}^2-1\right)}^2}=\left({\mathrm{a}}^2-1\right)$
Required area = 2a x (a² - 1)=  f(a)
f'(a) = 2a(2a - 0)+ (a² -1)2
$=4{\mathrm{a}}^2+2\left({\mathrm{a}}^2-1\right)=0\Rightarrow 6{\mathrm{a}}^2=2\Rightarrow {\mathrm{a}}^2=\frac{1}{3}$
Area $=\left|2\left(\frac{1}{\sqrt{3}}\right)\times \left(\frac{1}{3}-1\right)\right|=\frac{4}{3\sqrt{3}}$ square units.