Question
Let f_ 1 :(0, ) R and f_ 2 :(0, ) R be defined by f_ 1 (x)= _ 0 ^ x _ j=1 ^ 21 (t-j)^ j d t, x > 0 and f_ 2 (x)=98(x-1)^ 50 -600(x-1)^ 49 +2450, x > 0 where, for any positive integer n and real numbers a_ 1 , a_ 2 , , a_ n , _ i=1 ^ n a_ i denotes the product of a_ 1 , a_ 2 , , 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 value of 6 m 2 + 4 n 2 + 8 m 2 n 2 is ____.