Let S be the set of all twice differentiable functions $$f$$ from $$\mathbb{R}$$ to $$\mathbb{R}$$ such that $$\frac{d^2f}{dx^2}(x) > 0$$ 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?

Let $$f \in S$$.
Define $$g(x)=f(x)-x \;,\; x\in \mathbb{R}$$.
Then $$g''(x)=f''(x)\gt 0$$ for all $$x\in (-1,1)$$, so $$g$$ is a strictly convex function on $$(-1,1)$$.

A basic property of strictly convex functions is:
Property - A strictly convex function defined on an interval can intersect any straight line in at most two distinct points.
(If it had three or more intersections, the chord between the outer two points would lie strictly below the graph at the middle point, contradicting convexity.)

Here the line is $$y=0$$ for the function $$g(x)$$. Hence, inside $$(-1,1)$$ the equation $$g(x)=0$$ - equivalently $$f(x)=x$$ - can have at most two distinct solutions.

Therefore, for every $$f\in S$$, the number of fixed points satisfies $$X_f\le 2$$.
So Option B is TRUE.

Next we examine the existence of functions giving $$X_f=0,1,2$$.

Choose $$f_0(x)=x+x^2+1$$.
Then $$f_0''(x)=2\gt 0$$ for all $$x$$, so $$f_0\in S$$.
Moreover $$f_0(x)-x=x^2+1\gt 0$$ for every $$x\in (-1,1)$$, hence there is no solution of $$f_0(x)=x$$ inside that interval and $$X_{f_0}=0$$.
Thus Option A is TRUE.

Choose $$f_1(x)=x+x^2$$.
Again $$f_1''(x)=2\gt 0$$, so $$f_1\in S$$.
Now $$f_1(x)=x \;\Longleftrightarrow\; x^2=0 \;\Longleftrightarrow\; x=0$$, which lies inside $$(-1,1)$$. Hence $$X_{f_1}=1$$.
Because such a function exists, the statement “There does NOT exist any function with $$X_f=1$$” is false, so Option D is FALSE.

Choose $$f_2(x)=x+x^2-\dfrac14$$.
Here $$f_2''(x)=2\gt 0$$, so $$f_2\in S$$.
Solve $$f_2(x)=x$$: $$x+x^2-\dfrac14 = x \;\Longleftrightarrow\; x^2-\dfrac14 =0 \;\Longleftrightarrow\; x=\pm\dfrac12$$.
Both solutions lie in $$(-1,1)$$, giving $$X_{f_2}=2$$.
Thus Option C is TRUE.

Combining all the above, the correct statements are:
Option A, Option B and Option C.