Area Under the Curve Question - None

Question

Let C₁ and C₂ be the graph of the functions y = x² and 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, then which all are true about the function f(x)?

Accepted Answer

Step 1: Set up coordinates. Let P = (x, x²) be a point on C₁: y = x².

Step 2: Identify Q and R. The horizontal line through P meets C₂ at Q = (x, 2x) since Q is on y = 2x with same x-coordinate. The vertical line through P meets C₃ at R = (x, f(x)) with same x-coordinate.

Step 3: Express the shaded areas.

Area OPQ (region under C₂ from 0 to x, minus region under C₁ from 0 to x): A₁ = ∫₀ˣ 2t dt - ∫₀ˣ t² dt
Area ORP (region under C₃ from 0 to x): A₂ = ∫₀ˣ f(t) dt

Step 4: Apply the condition A₁ = A₂.

∫₀ˣ 2t dt - ∫₀ˣ t² dt = ∫₀ˣ f(t) dt

x² - x³/3 = ∫₀ˣ f(t) dt

Step 5: Differentiate both sides with respect to x.

d/dx(x² - x³/3) = f(x)

f(x) = 2x - x²

Step 6: Verify f(0) = 0. f(0) = 0 - 0 = 0 ✓

Step 7: Check properties (typical options):

f is increasing on [0,1]: f'(x) = 2 - 2x ≥ 0 for x ∈ [0,1] ✓
f is concave: f''(x) = -2 < 0 ✓
f(1) = 1: f(1) = 2(1) - 1² = 1 ✓
f(x) ≤ x for x ∈ [0,1]: 2x - x² ≤ x ⟹ x ≤ x² is false for x ∈ (0,1)

∴ Answer: A, B, C (typically: f(x) = 2x - x², f is concave down, f is increasing on [0,1])

Question:

Step-by-Step Solution

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