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Functions question

2023 · Shift 1 · Q20

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PYQ Library
JEE Advanced
Mathematics
Functions
2023 · Shift 1 · Q20

Functions question

2023 · Shift 1 · Q20

JEE AdvancedMathematicsFunctionsMultiple correct+4 / −2

Let

f:[0,1] \rightarrow[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 \lt 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?

A
There exists an $h \in\left[\frac{1}{4}, \frac{2}{3}\right]$ 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 $h \in\left[\frac{1}{4}, \frac{2}{3}\right]$ 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 $h \in\left[\frac{1}{4}, \frac{2}{3}\right]$ 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 $h \in\left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the green region below the line $L_k$

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