Let and be the real roots of the quadratic equation x^2 - (2 )x + ^2 = 0 , where R . If M and m are the maximum and minimum values of ^4 + ^4 respectively, then the value of M + 2m is :

Step-by-step solution

For the quadratic equation to have real roots, its discriminant must be non-negative: D = (-2 )^2 - 4(1)( ^2 ) 0 4 ^2 - 4 ^2 0 ^2 ^2 Since ^2 = 1 - ^2 , we get 2 ^2 1 ^2 1 2 . Let u = ^2 . The valid domain for u is [ 1 2 , 1 ] . From Vieta's formulas, the sum and product of the roots are: S = + = 2 P = = ^2 = 1 - u We need to find the extrema of ^4 + ^4 . First, find ^2 + ^2 : ^2 + ^2 = S^2 - 2P = 4 ^2 - 2 ^2 = 4u - 2(1-u) = 6u - 2 . Now, expand ^4 + ^4 : ^4 + ^4 = ( ^2 + ^2)^2 - 2P^2 = (6u - 2)^2 - 2(1 - u)^2 = (3