Assume for all $0<\alpha<1$ and $x \in [0,\infty)$, $\frac{\partial^3 \log f(x,\alpha)}{\partial \alpha^3}$ and $\frac{\partial^3 \log g(x,\alpha)}{\partial \alpha^3}$ exist, where $f(x,\alpha)$ and $g(x,\alpha)$ are two functions of $x$ and $\alpha$. Is the followings are true? Provide counter example or prove it.
A) $\frac{\partial^3 \log (f(x,\alpha)+g(x,\alpha))}{\partial \alpha^3}$ exist for all $0<\alpha<1$ and $x \in [0,\infty)$.
B) $\frac{\partial^3 \log (f(x,\alpha)*g(x,\alpha))}{\partial \alpha^3}$ exist for all $0<\alpha<1$ and $x \in [0,\infty)$, where $*$ denotes the convolution with respect to $x$.
My answer:
Intuitively, I think both of them are true but I have no idea how to prove. For part A, I thought of taking the derivatives and then see how the function will behave. But, that will not work since we first should know the derivative exists.
For the second, I think we should use this idea that $\frac{d h_1(x)*h_2(x)}{dx} = \frac{d h_1(x)}{dx}*h_2(x)=h_1(x)*\frac{d h_2(x)}{dx}$. I appreciate a clear answer to both of them.