Question
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) true?
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/minima, f ' x = 0 ⇒ x = 1 3 Now finding the area of red region we get, A R = ∫ 0 1 f x   d x = 1 2 So, area of the green region will be, A G = 1 2 , as total area is 1 sq.unit Now solving option A we get, 1 - h = h - 1 2 ⇒ h = 3 4 ,   3 4 > 2 3 So, option A is incorrect For option B h = 1 2 - h ⇒ h = 1 4 So, option B is correct as h ∈ 1 4 , 2 3 Now solving option C we get, ∫ 0 1 f x   d x = 1 2 , ∫ 0 1 1 2 d x = 1 2 ⇒ ∫ 0 1 f x - 1 2 d x = 0 ⇒ h = 1 2 So, option C is correct. D ∵ Option C is correct ⇒ option D is also correct.