Question

Let f(x) = maximum {x², (1 - x)², 2x (1 - x)} where 0 ≤ x ≤ 1. Determine the area of the region bounded by the curves y = f(x), x-axis, x = 0 and x = 1.

Answer 1

See Fig.

y = x² is parabola (C₁) with vertex (0, 0) and passing through B(1, 1), y = (1 - x)² is parabola (C₂) with vertex at A(1, 0) and passing through D (0, 1).

y = 2x(1 - x) = 2x - 2x² = - 2(x - 1/2)² + 1/2 is parabola with vertex at H(1/2, 1/2).