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the_ace
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1. Find the absolute maxima and the absolute minima of the following function
f(x)=(2x)/(x^2+1) on [-2,2]
f(x)=(2x)/(x^2+1) on [-2,2]
Absolute maximum and minimum refer to the highest and lowest values that a function can attain over a given interval. They represent the global extremes of the function and are often referred to as the overall maximum and minimum.
To find the absolute maximum and minimum of a function, you need to first find the critical points by taking the derivative of the function and setting it equal to zero. Then, evaluate the function at these points as well as at the endpoints of the given interval. The highest and lowest values obtained from these evaluations will be the absolute maximum and minimum of the function.
Local maxima and minima refer to the highest and lowest values within a small interval around a specific point on the function. They represent the highest or lowest point in that particular region but may not be the overall highest or lowest point of the function. Absolute maxima and minima, on the other hand, refer to the highest and lowest values of the entire function over a given interval.
Yes, a function can have multiple absolute maxima or minima. This can occur when the function has multiple peaks or valleys within the given interval. In this case, each peak and valley would represent an absolute maximum or minimum.
Finding the absolute maxima and minima of a function is important because it helps us understand the behavior of the function and identify its extreme values. This information can be useful in various applications, such as optimizing a process or determining the maximum or minimum value of a quantity.