Let _ and _ be the distinct roots of 2 x^2+( ) x-1=0, (0,2 ) . If m and M are the minimum and the maximum values of _ ^4+ _ ^4 , then 16(M+m) equals :

Step-by-step solution

aligned & 2 x^2+( ) x-1=0 & _ + _ = - 2 & _ _ = -1 2 & _ ^2+ _ ^2= ( _ + _ )^2-2 _ _ ^2 4 +1 aligned aligned & _ ^4+ _ ^4= (a_ ^2+ _ ^2 )^2-2 _ ^2 _ ^2= ( ^2 4 +1 )^2- 2 4 & = ( ^2 4 +1 )^2- 1 2 aligned Maximum when =1 aligned & M= ( 1 4 +1 )^2- 1 2 & M= 17 16 aligned Minimum when =0 aligned & m=1- 1 2 = 1 2 & 16(M+m)=16 ( 17 16 + 1 2 )=25 aligned