Let S be the set of all twice differentiable functions f from ℝ to ℝ such that ‌
d2f
dx2
(x)>0 for all x∈(−1,1). For f∈S, let Xf be the number of points x∈(−1,1) for which f(x)=x. Then which of the following statements is(are) true?
[JEE Adv 2023 P2]

Question

Let S be the set of all twice differentiable functions f from ℝ to ℝ such that ‌

d2f

dx2

(x)>0 for all x∈(−1,1). For f∈S, let Xf be the number of points x∈(−1,1) for which f(x)=x. Then which of the following statements is(are) true?

[JEE Adv 2023 P2]

Answer

There exists a function f∈S such that Xf=0

Answer

For every function f∈S, we have Xt≤2

Answer

There exists a function f∈S, such that Xt=2

Answer

There does NOT exist any function f in S such that Xf=1

JEE Advanced 2023 Paper 2 Solutions