Let Max 0 ⩽ x ⩽ 2 9 - x 2 5 - x = α and Min 0 ⩽ x ⩽ 2 9 - x 2 5 - x = β . If ∫ β - 8 3 2 α - 1 Max 9 - x 2 5 - x , x d x = α 1 + α 2 log e 8 15 , then α 1 + α 2 is equal to ______

Step-by-step solution

Here y = 9 - x 2 5 - x = 5 + x + 16 x - 5 d y d x = 1 - 16 x - 5 2 When d y d x = 0 ⇒ 1 - 16 x - 5 2 = 0 So critical point is x = 1 in 0 , 2 Now we get y 0 = 9 5 , y 1 = 2 , y 2 = 5 3 So α = 2 and β = 5 3 I = ∫ - 1 3 max 9 - x 2 5 - x , x I = ∫ - 1 9 5 5 + x + 16 x - 5 d x + ∫ 9 5 3 x d x = 5 x + x 2 2 + 16 ln x - 5 - 1 9 5 + x 2 2 9 5 3 I = 18 + 16 ln 8 15 α 1 = 18 and α 2 = 16 ⇒ α 1 + α 2 = 34