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Optimization problems are an application of derivatives in calculus that allow us to find the local and global extrema of a function, including the local and global minima and the local and global maxima.
In order to find the extrema of a function, you’ll need to take the derivative of the function, set it equal to zero, and solve it for the independent variable in order to find critical numbers. These critical numbers represent potential critical points, which are the points at which the function changes direction from increasing to decreasing, or vice versa.
Once you’ve found the critical numbers, you’ll need to use the first derivative test to test each interval between the critical numbers to see where the function is increasing and where it’s decreasing. If the function is increasing to the left of the critical number, and decreasing to the right of it, the critical number represents a local maximum, and possibly a global maximum. On the other hand, if the function is decreasing to the left of the critical number and increasing to the right of it, the critical number represents a local minimum, and possibly a global minimum.