A rectangle with its sides parallel to the X -axis and Y -axis is inscribed in the region bounded by the curves y = x 2 - 4 and 2 y = 4 - x 2 . The maximum possible area of such a rectangle is closest to the integer.

Step-by-step solution

Equation of parabola y = x 2 - 4 and 2 y = 4 - x 2 Let point B = h , 4 - h 2 2 A = - h , 4 - h 2 2 C = h 1 h 2 - 4 D = - h , h 2 - 4 Area of rectangle A B C D = A B × B C = 2 h × 4 - h 2 2 - h 2 + 4 ⇒ A = 12 h - 3 h 3 ⇒ d A d h = 12 - 9 h 2 For maxima or minima d A d h = 0 ⇒ 12 - 9 h 2 = 0 h = ± 2 3 maximum at h = 2 3 ∴ A = 12 2 3 - 3 2 3 3 = 24 3 - 8 3 = 16 3 = 16 3 3 A - 16 × 173 3 = 9 . 22 A = 9 . 22 = 9