Consider the square region S=[0, 1]×[0, 1].Let G={(x,y)∈S:y>f(x)} be called the GREEN regionR={(x,y)∈S:y<f(x)} be called the RED region. Lh={(x,h)∈S:x∈[0,1]} be the horizontal line drawn at a height h∈[0,1].If f:[0,1]→[0,1] be the function defined as f(x)=x33−x2+5x9+1136, then which of the following statements is( are) true ?

There exists an h ∈ [ 1 6 , 2 3 ] such that area of RED region above line L h equals that of RED region below the line L h . , There exists an h ∈ [ 1 6 , 2 3 ] such that area of GREEN region above line L h equals that of RED region below the line L h . , There exists an h ∈ [ 1 6 , 2 3 ] such that area of GREEN region above line L h equals that of GREEN region below the line L h . , There exists an h ∈ [ 1 6 , 2 3 ] such that area of RED region above line L h equals that of GREEN region below the line L h .

Consider the square region S=[0, 1]×[0, 1].Let G={(x,y)∈S:y>f(x)} be called the GREEN regionR={(x,y)∈S:y

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Consider the square region $S = [0, 1] \times [0, 1] .$
Let $G = {(x, y) \in S : y > f (x)}$ be called the GREEN region
$R = {(x, y) \in S : y < f (x)}$ be called the RED region.
$L_{h} = {(x, h) \in S : x \in [0, 1]}$ be the horizontal line drawn at a height $h \in [0, 1] .$
If $f : [0, 1] \to [0, 1]$ be the function defined as $f (x) = \frac{x^{3}}{3} - x^{2} + \frac{5 x}{9} + \frac{11}{36},$ then which of the following statements is( are) true ?

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There exists an $h \in [\frac{1}{6}, \frac{2}{3}]$ such that area of RED region above line $L_{h}$ equals that of RED region below the line $L_{h}$ .

There exists an $h \in [\frac{1}{6}, \frac{2}{3}]$ such that area of GREEN region above line $L_{h}$ equals that of RED region below the line $L_{h}$ .

There exists an $h \in [\frac{1}{6}, \frac{2}{3}]$ such that area of GREEN region above line $L_{h}$ equals that of GREEN region below the line $L_{h}$ .

There exists an $h \in [\frac{1}{6}, \frac{2}{3}]$ such that area of RED region above line $L_{h}$ equals that of GREEN region below the line $L_{h}$ .