The sum of the maximum and minimum values of the function f x = 5 x - 7 + x 2 + 2 x in the interval 5 4 , 2 , where t is the greatest integer ≤ t , is ______.

Step-by-step solution

Given, f x = 5 x - 7 + x 2 + 2 x = 5 x - 7 + x + 1 2 - 1 = 5 x - 7 + x + 1 2 - 1 as  x - 1 = x -1 where  .  is greatest integer function Now critical points of f x = 7 5 , 5 - 1 , 6 - 1 , 7 - 1 , 8 - 1 , 2 ∴ Maximum or minimum value of f x occur at critical points or boundary points. So,   f 5 4 = 3 4 + 4 = 19 4 and f 7 5 = 0 + 4 = 4 As both 5 x - 7 and x 2 + 2 x are increasing in nature after x = 7 5 So, f 2 = 10 - 7 + 4 + 4 = 3 + 8 = 11 ∴   f 7 5 min = 4 an