A critical point is a point on a graph where the derivative is equal to zero, or where the derivative does not exist. It is also known as a stationary point, as the slope of the graph is horizontal at this point.
Finding critical points is important because it helps us identify the maximum and minimum values of a function. These points can also provide information about the behavior of a function and can be used to optimize a system.
To find the critical points of a function, you need to take the derivative of the function and set it equal to zero. Then, solve for the variable to find the x-values of the critical points. These x-values can then be substituted back into the original function to find the corresponding y-values.
Relative critical points are points where the derivative is equal to zero, while absolute critical points are points where the derivative does not exist. Absolute critical points are also known as singular points, and they can be found by setting the derivative equal to undefined values, such as infinity or negative infinity.
To determine the type of critical point, you need to analyze the behavior of the function on either side of the critical point. If the function is increasing before the critical point and decreasing after, it is a maximum point. If the function is decreasing before the critical point and increasing after, it is a minimum point. If the function is increasing or decreasing on both sides of the critical point, it is a point of inflection.
