If the line x=α divides the area of region R={(x,y)∈R²: x³≤y≤x, 0≤x≤1} into two equal parts, then

Area(x=0→x=α)=Area of ΔOAB - Area of ΔOCB = 1/2 α⋅α - ∫α₀x²dx = α²/2 - α³/4
Area(0=α→x=1)=Area of □BAEF - Area of □BCEF
1/2(α+1)(1-α) - ∫₁αx³ dx
1 - α²/2 - (1/4 - α⁴/4)
According to the question:
α²/2 - α³/4 = 1/2 - α²/2 + α⁴/4
2α⁴ - α² + 1 = 0
Option B
f(α) = 2α⁴ - α² + 1 = 0
At α=0, f(α) = 1
At α=1, f(α) = 2
At α=1/2, f(α) = 1/8 > 0
Therefore, Root lies in α∈(1/2,1).
Option D