Extremum Calculation An extremum is the point in the domain of a function where the function achieves its greatest or least value. Finding the extremum of a...
Extremum Calculation
An extremum is the point in the domain of a function where the function achieves its greatest or least value. Finding the extremum of a function is crucial in various applications, such as finding the maximum or minimum values of a physical quantity, the point of maximum curvature for a curve, and the point of minimum risk in financial investments.
To calculate the extremum, we need to find the points where the first derivative is equal to zero. These points are called critical points. Then, we evaluate the second derivative at each critical point to determine the nature of the extremum.
If the second derivative is positive, the critical point is a local minimum.
If the second derivative is negative, the critical point is a local maximum.
If the second derivative is zero, the critical point can be a local maximum, minimum, or saddle point.
In addition to finding the critical points, we can also use graphical techniques and optimization methods to identify the absolute maximum and minimum values of the function