For all real numbers x, the minimum value of 1 + 2cos(4x) is Solution The range of values of cos(4x) is as follows - 1 = cos(4x) = 1 Multiply all terms of the above inequality by 2 to obtain - 2 = 2 cos(4x) = 2 Add +1 to all terms of the inequality and simplify - 2 + 1 = 2 cos(4x) + 1 = 2 + 1 -1 = 2 cos(4x) + 1 = 3 The minimum value of 1 + 2cos(4x) is - 1.