Good work but you forgot the importance of checking the second derivative. It has to be non zero for your first derivate zeros to be the max/min of the original function.
They also tell you (if they are non zero) whether your original has a minimum or maximum. If the second derivative is positive it means your first derivative is transfering from negative to positive at its zero and the original function first had a decline in values and after that rises in values which means it's a minimum. Vice versa for the maximum.

Should both derivatives be zero you have a saddle point in your original function.