Let f : [0, 1] → [0, 1] be the function defined by Consider the square region S = [0, 1]…

Mathematics · JEE Advanced · NTA Exams — Limit, Continuity and Differentiability

Let f : [0, 1] → [0, 1] be the function defined byConsider the square region S = [0, 1] × [0, 1]. Let G = {(x, y) ∈ S : y > f(x)} be called the green region and R = {(x, y) ∈ S : y < f(x)} be called the red region. Let L_h = {(x, h) ∈ S : x ∈[0, 1]} be the horizontal line drawn at a height h ∈ [0, 1]. Then which of the following statements is (are) true?

1. There exists an such that the area of the green region above the line L_h equals the area of the green region below the line L_h
2. There exists an such that the area of the red region above the line L_h equals the area of the red region below the line L_h
3. There exists an such that the area of the green region above the line L_h equals the area of the red region below the line L_h
4. There exists an such that the area of the red region above the line L_h equals the area of the green region below the line L_h