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

Question

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 (A) (4/3 √3) (B) (1/3 √3) (C) (4/3) (D) (2/3 √3)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/e58fb643b54cb6568312ebf55f1f28c3-.png" /> Area ( A )=2 t ⋅(1- t 2) (0< t <1) A =2 t -2 t 3 ( d A / dt )=2-6 t 2 t =(1/√3) ⇒ A max =(2/√3)(1-(1/3))=(4/3 √3)

Q. The area (in sq. units) of the largest rectangle $A BC D$ 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

$\frac{4}{3 3}$

$\frac{1}{3 3}$

$\frac{4}{3}$

$\frac{2}{3 3}$

Area $(A) = 2 t \cdot (1 - t^{2})$
$(0 < t < 1)$
$A = 2 t - 2 t^{3}$
$\frac{d A}{d t} = 2 - 6 t^{2}$
$t = \frac{1}{3}$
$\Rightarrow A_{m a x} = \frac{2}{3} (1 - \frac{1}{3}) = \frac{4}{3 3}$

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=x2−1 below the x -axis, is

33​4​

33​1​

34​

33​2​