Question

Let C₁ and C₂ be the graphs of the functions y = x², y = 2x, 0 ≤ x ≤ 1 respectively. Let C₃ be the graph of a function y = f(x), 0 ≤ x ≤ 1, f(0) = 0.

For a point P on C₁, let the lines through P parallel to the axes meet C₂ and C₃ at Q and R, respectively. If for every position of P on C₁, the area of the shaded regions OPQ and ORP are equal, determine the function f(x).

Answer 1

Let C₁ and C₂ be the graphs of dunctions y = x², y = 2x, 0 ≤ x ≤ 1 respectively. Let C₃ be the graph of a function 

y = f(x), 0 ≤ x ≤ 1, f(0) = 0.

Let the co-ordinates P be (x, x²), where 0 ≤ x ≤ 1.

Area (ORPO) 

According to the question,

Differentiating both sides w.r.t. x, we get

x² – f(x) = 2x² – x³

f(x) = x³ – x²

Hence, the required curve is

f(x) = x³ – x², where 0 ≤ x ≤ 1