If $f$ is twice continously differentiable in (a,b) and if for all x $\in (a,b)$ $f''(x) + 2f'(x) + 3f(x) = 0$, the $f$ is infinitely differentiable on (a,b)
Let $f \in C[a,b]$ be a differentiable in (a,b). If $f(a) = f(b) = 0$, then for any real number $\alpha$, there exist x $\in$ (a,b) such that $f'(x) + \alpha f(x) = 0 $
The function defined below is not differentiable at x = 0 $$ \begin{equation} f(x)=\begin{cases} x^2|cos\frac{\pi}{x}|, & \text{ x $\neq 0$}\\ 0, & \text{x = 0 }. \end{cases} \end{equation}$$
For (1) and (2) are true , but i am not sure
For (3) is false
Thank you for sparing your valuable time in checking my solutions