Let f₁:(0, ) R and f₂:(0, ) R be defined by f₁(x)= ₀^ x _ j=1 ²¹(t-j)^ j d t, x>0 and f₂(x)=98(x-1)⁵⁰-600(x-1)⁴⁹+2450, x>0 where, for any positive integer n and real numbers a₁, a₂, , a_ n , _ i=1 ^ n a_ i denotes the product of a₁, a₂, , a_ n . Let m_ i and n_ i , respectively, denote the number of points of local minima and the number of points of local maxima of function f_ i , i=1,2 , in the interval (0, ) . The

Step-by-step solution

Given f 1 x = ∫ 0 x t − 1 t − 2 2 ...... t − 2 t 21 d t ⇒       f ' 1 x = x − 1 x − 2 2 ..... x − 21 21 Checking critical points At all odd integers from 1 to 21 f x will have an extrema with 1 , 5 , 9 , 13 , 17 , 21 being points of minima & 3 , 7 , 11 , 15 , 19 being points of maxima. So m 1 = 6 & n 1 = 5 Hence 2 m 1 + 3 n 1 + m 1 n 1 = 57