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

Given,

${f}^{"} ( x ) > 0 \text{\text{ and } }$ f ( x ) - x = 0

Now let,

$g\left(x\right) = f\left(x\right) - x$

Now differentiating above equation we get,

$⇒ {g}^{'}\left(x\right) = {f}^{'}\left(x\right) - 1$

Again differentiating we get,

${g}^{"} ( x ) = {f}^{"} ( x ) > 0 ⇒ \text{concave up}$

So, possible graphs will be,

Here in above graph ${X}_{f} = 0$ as there is no point for which $f\left(x\right) = x$

Now here ${X}_{f} \le 2$ as there will be minimum two points when $y = x$ will cut the given graph,

Now in above graph ${X}_{f} = 1$ so there will minimum one intersection between line $y = x$ and the given graph so option $\left(D\right)$ is wrong.