JEE Main · 2020 · 04 Sep 2020 · Shift 2 · Application of Derivatives

The area (in sq. units) of the largest rectangle

A B C D

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:

Answer: (A)

Solution

By symmetry take

A = (α, 0)

and

B = (- α, 0)

on the

x

-axis, with

C

and

D

y = x^{2} - 1

below it, so the rectangle has width

2 α

and height

1 - α^{2}

and area

S = 2 α (1 - α^{2}) = 2 α - 2 α^{3}

. Setting

\frac{d S}{d α} = 2 - 6 α^{2} = 0

gives

α = \frac{1}{3}

, and the largest area is

2 \cdot \frac{1}{3} - 2 \cdot \frac{1}{3 3} = \frac{4}{3 3}

JEE Main · 2020 · 04 Sep 2020 · Shift 2 · Single correct option