Let M and m be the absolute maximum and absolute minimum values of the function f(x) = 12( x + x) 2 + 2x on the interval [0, 2 ] . The value of M + m 2 is equal to:

Step-by-step solution

Let t = x + x . Squaring both sides, we get t^2 = ^2 x + ^2 x + 2 x x = 1 + 2x . Therefore, 2x = t^2 - 1 . The given function can be rewritten in terms of t as: g(t) = 12t 2 + (t^2 - 1) = 12t t^2 + 1 Now, we find the range of t for x [0, 2 ] . t = x + x = 2 (x + 4 ) Since 0 x 2 , we have 4 x + 4 3 4 . In this interval, 1 2 (x + 4 ) 1 . Thus, 1 t 2 . To find the extrema of g(t) on [1, 2 ] , we differentiate g(t) with respect to t : g'(t) = 12 ( (t^2 + 1)(1) - t(2t) (t^2 + 1)^2 ) = 12 ( 1 - t^2 (t^2 + 1)^2 ) For t (1