Which of the folowing is true (NBHM-2014) 
*

*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

HINT:
 For (b), you can see that the solution is:
 $f(x)=A e^{-\alpha x}$, where  $A$ being arbitrary constant.
Now $f(a)=f(b)=0$ gives $A=0$ which implies $f(x)=0$ is the solution for any $\alpha\in (a,b)$.
