Let \(f\) be a twice differentiable function defined on \(\mathbb{R}\) such that \(f(0)=1\), \(f'(0)=2\) and \(f'(x)\neq 0\) for all \(x\in \mathbb{R}\). If \(\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix}=0\), for all \(x\in \mathbb{R}\), then the value of \(f(1)\) lies in the interval:
