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

Options

A. 10
B. 9
C. 8
D. 7

Answer

B

Explanation

`y=x^(2)-4 & 2y=4-x^(2)`
`P(alpha,beta) implies 2 beta=4-alpha^(2)`
`P(alpha,(4-alpha^(2))/(2))`
`Q(-alpha,(4-alpha^(2))/(2))`
`PQ= 2 alpha`
`S(alpha,alpha^(2)-4),R(-alpha,-4+alpha^(2))`
`PS=(alpha^(2)-4-(4-alpha^(2))/(2))=(3alpha^(2)-12)/(2)`

`Area=PQPS=2alpha.(3alpha^(2)-12)/(2)=3alpha^(3)-12alpha=A(alpha)`
`A'(alpha)=9alpha^(2)-12=0impliesalpha^(2)(12)/(9)=(4)/(3)`
`alpha= pm (2)/(sqrt(3))` maximum at `alpha= -(2)/(sqrt(3))`
Maix. `=A(-(2)/(Sqrt(3)))=-3. (8)/(3sqrt(3))+12.(2)/(sqrt(3))=(16)/(sqrt(3))=(16sqrt(3))/(3)=9.22~~9`