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JEE Advanced 2023 · Paper 2 · one or more correct

JEE Advanced 2023 Mathematics question on Application of Derivatives

Let SS be the set of all twice differentiable functions ff from R\mathbb{R} to R\mathbb{R} such that d2fdx2(x)>0\dfrac{d^2f}{dx^2}(x) > 0 for all x∈(−1,1)x \in (-1,1). For f∈Sf \in S, let XfX_f be the number of points x∈(−1,1)x \in (-1,1) for which f(x)=xf(x) = x. Then which of the following statements is(are) true?
  1. A.
    There exists a function f∈Sf \in S such that Xf=0X_f = 0
  2. B.
    For every function f∈Sf \in S, we have Xf≤2X_f \leq 2
  3. C.
    There exists a function f∈Sf \in S such that Xf=2X_f = 2
  4. D.
    There does NOT exist any function ff in SS such that Xf=1X_f = 1
Show answer

Correct options: A, B, C

[JEE Advanced 2023 · Paper 2]

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