Let and be the roots of the quadratic equation 2x^2 - ( + )x - 1 = 0 , where R . If M and m are the maximum and minimum values of the expression ^2 + ^2 respectively, then the value of M^2 + m^2 is :

Step-by-step solution

Let S = + = + 2 and P = = - 1 2 . The given expression is ^2 + ^2 = ^3+ ^3 ^2 ^2 . Using the identity ^3+ ^3 = ( + )^3 - 3 ( + ) , we get: S^3 - 3PS P^2 = S^3 - 3 (- 1 2 )S (- 1 2 )^2 = 4 (S^3 + 3 2 S ) = 4S^3 + 6S . Let t = + . We know that the range of t is [- 2 , 2 ] . Substituting S = t 2 , the expression becomes: f(t) = 4 ( t 2 )^3 + 6 ( t 2 ) = t^3 2 + 3t . To find the extrema, we differentiate f(t) with respect to t : f'(t) = 3 2 t^2 + 3 . Since f'(t) > 0 for all real t , f(t) is a strictly increasing functi