Let A = a i j be a 3 × 3 matrix, where a i j = 1 , if i = j - x , if i - j = 1 2 x + 1 , otherwise Let a function f : R → R be defined as f x = det A . Then the sum of maximum and minimum values of f on R is equal to:

Step-by-step solution

We have, a i j = 1 ,   if   i = j - x ,   if   i - j = 1 2 x + 1 ,   otherwise   Then, A = 1 − x 2 x + 1 − x 1 − x 2 x + 1 − x 1 ⇒     A = 1 - x 2 + x - x + 2 x 2 + x + 2 x + 1 x 2 - 2 x - 1 ⇒ A = 1 - x 2 + 2 x 3 + 2 x 3 - 4 x 2 - 2 x + x 2 - 2 x - 1 ⇒ A = f x = 4 x 3 - 4 x 2 - 4 x ⇒ f ' x = 4 3 x 2 − 2 x − 1 For maxima or minima, f ' x = 0 ⇒     3 x 2 − 2 x − 1 = 0 ⇒   &#16