What are critical points?
Critical points are one of the best things we can do with derivatives, because critical points are the foundation of the optimization process. Optimization is all about finding the maxima and minima of a function, which are the points where the function reaches its largest and smallest values. And if we know which values maximize function and which ones minimize it, that has all kinds of real-world applications.
In order to find critical points, which are the points where the function might have extrema, we just take the derivative of the original function, set that derivative equal to 0, and then solve that equation for x. That gives us the critical points of the function. Then, we can test those critical points to determine whether they represent maxima, minima, or neither.