Let f: [0, 1] → [0, 1] be the function defined by x^3 /3 - x^2 + 5/9 x + 17/36.

Question

Let f: [0, 1] → [0, 1] be the function defined by \(\frac{x^3}3 - x^2 + \frac 59 x +\frac{17}{36}\). Consider 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?

(A) There exists an h = \(\left[\frac 14, \frac 23\right]\) 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

(B) There exists an h = \(\left[\frac 14, \frac 23\right]\) 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

(C) There exists an h = \(\left[\frac 14, \frac 23\right]\) 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

(D) There exists an h = \(\left[\frac 14, \frac 23\right]\) 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

Answer 1

Correct options are (B), (C) & (D)