Consider the functions f , g : ℝ → ℝ defined by f x = x 2 + 5 12 and g x = 2 1 - 4 x 3 , x ≤ 3 4 , 0 , x > 3 4 If α is the area of the region x , y ∈ ℝ × ℝ : x ≤ 3 4 , 0 ≤ y ≤ min f x , g x , then the value of 9 α is _____ .

Step-by-step solution

Plotting the diagram of the given function we have, Now by diagram we can see twice of A 1 and A 2 will be the required area, So, required area = 2 · A 1 + A 2 ⇒ α = ∫ 0 1 2 x 2 + 5 12 d x + 1 2 3 4 - 1 2 · 2 3 × 2 ⇒ α = x 3 3 + 5 x 12 0 1 2 + 1 2 ⋅ 1 4 ⋅ 2 3 × 2 ⇒    α = 1 4 + 1 12 × 2 ⇒    α = 2 3 ⇒ 9 α = 6