Let M_n be the absolute minimum value of the function f_n(x) = x^ n+2 - x^n on the interval [0, 1] , where n N . Then _ n (n M_n) is equal to

Step-by-step solution

f_n'(x) = (n+2)x^ n+1 - n x^ n-1 = x^ n-1 ((n+2)x^2 - n) . Setting f_n'(x) = 0 for x [0, 1] gives the critical point x^2 = n n+2 , so x = n n+2 . The minimum value M_n is obtained by substituting this into f_n(x) = x^n(x^2 - 1) : M_n = ( n n+2 )^n ( n n+2 - 1 ) = ( n n+2 )^ n 2 ( -2 n+2 ) . We need to evaluate _ n (n M_n) : _ n n M_n = _ n [ ( n n+2 )^ n 2 ( -2n n+2 ) ] . This can be split into two limits: _ n ( -2n n+2 ) = -2 . And _ n ( n n+2 )^ n 2 = _ n (1 - 2 n+2 )^ n 2 . This is a 1^ indeterminate form. The l