For a,c in R>0, define $f(x)=0$ for $x=0$ and $f(x)=x^a.sin(\frac{1}{|x|^c})$ for $x\not=0$. Relationship between a & c for f''(0) to exist?

Question

I thought of it as "Basically, what`s the relationship between a and c for the 2nd derivative of f(x) to exist at x=0?" I tried the primary approach and just differentiated the function twice, evaluated it at x=0, and got a mess of infinities and 0's.
Obviously, even though I do not know the answer, I am pretty sure that I am not doing it right, because it is blowing up.

Can any kind soul please help me with this?

Thank you so much!