Condition for absolute (global) max or min
The condition for finding the absolute (global) maximum or minimum of a function occurs at critical points and endpoints within a closed interval
The condition for finding the absolute (global) maximum or minimum of a function occurs at critical points and endpoints within a closed interval.
1. Critical Points: These are the points where the derivative of the function is either zero or undefined. To find critical points, you need to solve the equation f'(x) = 0 or f'(x) undefined. Once you have the critical points, you evaluate the function at these points and determine the corresponding values.
2. Endpoints: If the function is defined over a closed interval, you must also check the values of the function at the endpoints of the interval. Calculate the value of the function at the lower and upper bounds of the interval.
After obtaining the values of the function at both the critical points and the endpoints, compare these values to identify the absolute maximum and minimum values. The highest value will be the absolute maximum, and the lowest value will be the absolute minimum.
However, keep in mind that not all functions will have both an absolute maximum and minimum. Some functions may only have one of these values or neither in certain cases.
Additionally, it is crucial to note that the above condition assumes that the function is continuous within the closed interval being considered. If the function is not continuous, the concept of absolute maxima or minima may not apply.
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