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? (A) There exists a function $f \in S$ such that $X _ { f } = 0$ (B) For every function $f \in S$, we have $X _ { f } \leq 2$ (C) There exists a function $f \in S$ such that $X _ { f } = 2$ (D) There does NOT exist any function $f$ in $S$ such that $X _ { f } = 1$
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?
(A) There exists a function $f \in S$ such that $X _ { f } = 0$
(B) For every function $f \in S$, we have $X _ { f } \leq 2$
(C) There exists a function $f \in S$ such that $X _ { f } = 2$
(D) There does NOT exist any function $f$ in $S$ such that $X _ { f } = 1$