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
A
43√3
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B
43
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C
23√3
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D
13√3
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Solution
The correct option is A43√3
Area =2a(a2−1)
(Say) A=2a3−2a ⇒dAda=6a2−2
For maximum and minimum, dAda=0 ∴6a2−2=0 ⇒a=±1√3
Now, d2Ada2=12a
At a=1√3 ⇒d2Ada2=4√3>0 ∴a=1√3 (rejected)
At a=−1√3 ⇒d2Ada2=−4√3<0 ∴ Maximum area =(2a3−2a)=(−23√3+2√3) =43√3 sq. units