Let f:[0,1] \to[0,1] be the function defined by f(x)=\frac{x^3}{3}-x^2+\frac{5}{9} x+\frac{17}{36}. Consider the square region S=[0,1] \times[0,1]. Let G=\{(x, y) \in S: y>f(x)\} be called the green region and R=\{(x, y) \in S: y< f(x)\} be called the red region. Let L_h=\{(x, h) \in S: x \in[0,1]\} be the horizontal line drawn at a height h \in[0,1]. Then which of the following statements is(are) true? [JEE Adv 2023 P1]

There exists an h \in [\frac{1}{4}, \frac{2}{3}] 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

There exists an h \in [\frac{1}{4}, \frac{2}{3}] 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

There exists an h \in [\frac{1}{4}, \frac{2}{3}] 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

There exists an h \in [\frac{1}{4}, \frac{2}{3}] 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

Functions

Section: Mathematics

Question : 8 of 30

Marks: +1, -0

Let f:[0,1]→[0,1] be the function defined by f(x)=‌

−x2+‌

x+‌

. 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 Lh={(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?

[JEE Adv 2023 P1]

There exists an h∈[‌
1
4
,‌
2
3
] such that the area of the green region above the line Lh equals the area of the green region below the line Lh
There exists an h∈[‌
1
4
,‌
2
3
] such that the area of the red region above the line Lh equals the area of the red region below the line Lh
There exists an h∈[‌
1
4
,‌
2
3
] such that the area of the green region above the line Lh equals the area of the red region below the line Lh
There exists an h∈[‌
1
4
,‌
2
3
] such that the area of the red region above the line Lh equals the area of the green region below the line Lh

Validate

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