Let f : 1 ,   3 → R , be a function defined by f x = x x 1 + x 2 , where x , denotes the greatest integer ≤ x . Then the range of f , is

Step-by-step solution

f   : 1 , 3 → R ,   f x = x x 1 + x 2 f x = x 1 + x 2 , x ∈ 1 ,   2 2 x 1 + x 2 , x ∈ [ 2 ,   3 ) f ' x = 1 + x 2 1 - x 2 x 1 + x 2 2 , x ∈ 1 ,   2 1 + x 2 2 - 2 x 2 x 1 + x 2 2 , x ∈ [ 2 , 3 ) f ' x = 1 - x 2 1 + x 2 , x ∈ 1 ,   2 1 - 2 x 2 1 + x 2 , x ∈ [ 2 , 3 ) ∴   f x is decreasing function. ∴   R f   ∈ 2 5 ,   1 2 ∪ 6 10 ,   4 5