A critical point of a parametrized curve (x(t), y(t)) in the plane is a value t0 of t such that the tangent line is vertical or horizontal.
t0 is a critical point if and only if x(t) is defined and x'(t) = 0 or y(t) is defined and y'(t) = 0 (or both).
In this demo, increasing x is indicated by red, decreasing x by absence of red, increasing y by blue, and decreasing y by absence of blue. Consequently, when x and y are both increasing, the red and blue will overlap in a purple section of the curve. The curve will be white where both x and y are decreasing.
You can use these colors to find critical points. Any point at which the curve changes from red to white or purple (or vice versa) is a critical point for x(t) and any point at which the curve changes from blue to white or purple (or vice versa) is a critical point for y(t).