JEE Advanced2023MathematicsArea Under CurvesActual
Let f : 0 , 1 → 0 , 1 be the function defined by f x = x 3 3 - x 2 + 5 9 x + 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
Options
- AThere exists an h ∈ 1 4 ,   2 3 such that the area of the green region above the line L h equals th
- BThere exists an h ∈ 1 4 ,   2 3 such that the area of the red region above the line L h equals the
- CThere exists an h ∈ 1 4 ,   2 3 such that the area of the green region above the line L h equals th
- DThere exists an h ∈ 1 4 ,   2 3 such that the area of the red region above the line L h equals the
Correct answer
B. There exists an h ∈ 1 4 ,   2 3 such that the area of the red region above the line L h equals the
Step-by-step solution
Given, Function f x = x 3 3 - x 2 + 5 9 x + 17 36 Square region S = 0 ,   1 × 0 ,   1 Green region given by G = x ,   y ∈ S : y > f x Red region is given by R = x ,   y ∈ S : y < f x And L h = x ,   h ∈ S : x ∈ 0 ,   1 be the horizontal line drawn at a height h ∈ 0 ,   1 Now plotting the diagram of the above function we get, Now differentiating the function, f x = x 3 3 - x 2 + 5 9 x + 17 36 ⇒ f ' x = x 2 - 2 x + 5 9 For maxima/