Let \(\mathrm{f}:[0,1]\rightarrow \mathrm{R}\) (the set of all real numbers) be a function. Suppose the function \(\mathrm{f}\) is twice differentiable, \(\mathrm{f}(0)= \mathrm{f}(1)=0\) and satisfies \(\mathrm{f}''(\mathrm{x})-2\mathrm{f}'(\mathrm{x})+\mathrm{f}(\mathrm{x})\geq \mathrm{e}^{\mathrm{x}},\ \mathrm{x}\in[0,1]\).
Which of the following is true for \(0<\mathrm{x}<1\)?
1
\( 0<\mathrm{f}(\mathrm{x})<\infty\)
2
\(-\displaystyle \frac{1}{2}<\mathrm{f}(\mathrm{x})<\frac{1}{2}\)
3
\(-\displaystyle \frac{1}{4}<\mathrm{f}(\mathrm{x})<1\)
4
\(-\infty<\mathrm{f}(\mathrm{x})<0\)