Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{d^2 f}{d x^2}(x)>0$ for all $x \in(-1,1)$. For $f \in S$, let $X_f$ be the number of points $x \in(-1,1)$ for which $f(x)=x$. Then which of the following statements is(are) true?
Select ALL correct options:
A
There exists a function $\mathrm{f} \in \mathrm{S}$ such that $\mathrm{X}_{\mathrm{t}}=0$
B
For every function $\mathrm{f} \in \mathrm{S}$, we have $\mathrm{X}_{\mathrm{t}} \leq 2$
C
There exists a function $\mathrm{f} \in \mathrm{S}$ such that $\mathrm{X}_{\mathrm{t}}=2$
D
There does NOT exist any function $f$ in S such that $\mathrm{X}_1=1$
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