I think in general,

points of inflection occur where there is a concavity change, which means the 2nd derivative passes through zero.

critical points occur where the first derivative is zero or infinity/undefined

So for instance, on #1, take two derivatives with respect to x and then set it equal to zero and use the quadratic formula to solve for x.

For two variables, to find critical points you take an x derivative of the original function, and then a y derivative of the original function, and set both those derivatives to zero and solve for x and y (you have 2 equations with 2 unknowns).