Let S be the set of all twice differentiable functions f from ℝ to ℝ such that d 2 f d x 2 x > 0 for all x ∈ - 1 , 1 . For f ∈ S , let X f be the number of points x ∈ - 1 , 1 for which f x = x . Then which of the following statements is(are) true?
C. There exists a function f ∈ S , such that X f = 2.
Given, f " ( x ) > 0     &     f ( x ) - x = 0 Now let, g x = f x − x Now differentiating above equation we get, ⇒ g ' x = f ' x − 1 Again differentiating we get, g " ( x ) = f " ( x ) > 0 ⇒  concave up  So, possible graphs will be, Here in above graph X f = 0 as there is no point for which f x = x Now here X f ≤ 2 as there will be minimum two points when y = x will cut the given graph, Now in above graph X f = 1 so there will minimum one intersection between line y = x and the given graph so option D is wrong.
Related: Mathematics — Application of Derivatives · All PYQ Banks