Let y = f ( x ) be a thrice differentiable function in ( - 5 , 5 ) . Let the tangents to the curve y = f ( x ) at ( 1 , f ( 1 ) ) and ( 3 , f ( 3 ) ) make angles π 6 and π 4 , respectively with positive x -axis. If 27 ∫ 1 3 f ' t 2 + 1 f " t d t = α + β 3 where α , β are integers, then the value of α + β equals

Step-by-step solution

Given: y = f x ⇒ d y d x = f ' x d y d x 1 , f 1 = f ' 1 = tan π 6 = 1 3 ⇒ f ' 1 = 1 3 . . . i dy dx 3 , f 3 = f ' 3 = tan π 4 = 1 ⇒ f ' 3 = 1 . . . i i It is given that, 27 ∫ 1 3 f ' t 2 + 1 f " t d t = α + β 3 Let, I = ∫ 1 3 f ' ( t ) 2 + 1 f " t d t Putting, f ' ( t ) = z ⇒ f " t d t = d z z = f ' ( 3 ) = 1 z = f ' 1 = 1 3 ⇒ I = ∫ 1 3 1 z 2 + 1 d z ⇒ I = z 3 3 + z 1 3 1 ⇒ I = 1 3 + 1 - 1 3 · 1 3 3 + 1 3 ⇒ I = 4 3 - 10 9 3 ⇒ I = 4 3 - 10 27 3 ⇒ α + β 3 = 27 4 3 - 10 27 3 ⇒ α + β 3 = 36 - 10 3 ⇒ α = 36 , β = - 10