Let f x = sin ⁡ π x x 2 , x > 0 . Let x 1 < x 2 < x 3 < … < x n < … be all the points of local maximum of f and y 1 < y 2 < y 3 < … < y n < … be all the points of local minimum of f . Then which of the following options is/are correct?

Step-by-step solution

f ' x = 2 x cos ⁡ π x π x 2 - tan ⁡ π x x 4 …(i) ⇒ for maxima/minima f ' x = 0 ⇒ cos ⁡ π x = 0 o r π x 2 = tan ⁡ π x ∵ cos ⁡ π x ≠ 0 ∵ tan ⁡ π x will not be defined ∴ maxima/minima will occur Where tan ⁡ π x = π x 2 Case I : i For example In x ∈ 2 n , 2 n + 1 2 at point P 2 , x ∈ 2 , 5 2 cos ⁡ π x > 0 and at P 2 + , tan ⁡ π x > π x 2 at P 2 - , tan ⁡ π x π x 2 hence from equation (i) in x ∈ 2 n , 2 n + 1 2 , f ' x goes from positive to negative hence P 2 is maxima Similarly P 2 , P 4 , P 6 … are point of maxima and