Q: What are the common critical control points?

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The "critical points" of a function are the points at which the derivative equals zero or the derivative is undefined. To find the critical points, you first find the derivative of the function. You then set that derivative equal to zero. Any values at which the derivative equals zero are "critical points". You then determine if the derivative is ever undefined at a point (for example, because the denominator of a fraction is equal to zero at that point). Any such points are also called "critical points". In essence, the critical points are the relative minima or maxima of a function (where the graph of the function reverses direction) and can be easily determined by visually examining the graph.

If the two lines are actually "on top of each other", they can have infinitely many points in common. If they are parallel, they have no points in common. If they are perpendicular, they have one point in common.

Yes, adjacent angles do have common interior points.

If the denominator is zero at some point, then the function is not defined at the corresponding points.

Point common to two sides of an angle = vertex. Points common to two sides of a polygon, if they exist, are all the points along an edge.

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Take the derivative of the function and set it equal to zero. The solution(s) are your critical points.