If a α is the greatest term in the sequence a n = n 3 n 4 + 147 , n = 1 , 2 , 3 . . . . , then α is equal to _ _ _ _ _ _

Step-by-step solution

Let, y = x 3 x 4 + 147 = f x We know that, for increasing function d y d x > 0 So, differentiating the given function f x = x 3 x 4 + 147 we get, d y d x = - x 2 x 4 - 441 ( x 4 + 147 ) 2 ⇒ - x 2 x 4 - 441 ( x 4 + 147 ) 2 > 0 ⇒ x 2 x 4 - 441 ( x 4 + 147 ) 2 < 0 ⇒ x 4 < 441 Now for maxima/minima d y d x = 0 ⇒ x 4 = 441 ⇒ x = α ,   4 < α < 5 ⇒ Maximum value of f x is at x = 4 or x = 5 f 4 = 64 403 ,   f 5 = 125 772 ∵ f 5 > f 4 &