Question

what is the concavity and inflection points when the first derivative is 4x^3 + 100x?

Answer 1

Take the second derivative:
\[f'(x)=4x^3+100x \implies f''(x)=12x^2+100=0;\]
But that function is not zero for all (real valued) x's. So that means there are no points of inflection. So the concavity is the same everywhere.
Simply plug in x=0 (based solely off of convenience) to get f''(0)=100 >0 implies CCU
http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E4%2B50x%5E2
You can see from this (here I have integrated f'(x) to get f(x)).