Question

f(0) = 2, f '(x) > 0 for all x,
f '(0) = 1, f "(x) > 0 for x > 0,
f "(x) < 0 for x < 0, f "(0) = 0

How would I find out how the graph looks with these properties?

Answer 1

f(0)=2 is a point just plot (0,2)
f'(x)>0 means that the first derivative is positive therefore it is increasing
the other information is telling you about the 2nd derivative.
These are open ended which means answers can vary as long as all of the properties are correct.