A problem about differentiability

[rasi]

as you know i have been asked a question which no no way i couldn't tackle it. and its is about differentiabilty. at long last i found a solution. i want to share with you. could you check out please. thanks for now.this is the question.

and this is my solution.(i assume that when x goes infinity, the limit value of functions equals zero.)
part1

part2

 

[fresh_42]

I cannot decode the first picture. However, there are functions which locally violate the assertion. So the problem comes down to: If $f(x_0)^2 \leq 1$ and $f'(x_0)^2 +f''(x_0)\leq 1$ and $f(x_0)^2 +f'(x_0)^2> 1$, then there is no twice differentiable continuation of $f$ on $\mathbb{R}$. This means on the other hand, that the continuity of $f''(x)$ is crucial for the proof!