Absolute Minimum Value
Let f be a function with domain D. Then f(c) is the absolute maximum value on D if and only if f(x) ≥ f(c) for all x in D.
The absolute minimum value of a function refers to the smallest output value that the function can produce over its entire domain. If the function is continuous and defined on a closed interval, then the absolute minimum value is guaranteed to exist.
To find the absolute minimum value of a function, one can use various methods including:
1. Analytic Method – this involves taking the derivative of the function, finding the critical points, and evaluating the function at those points as well as at the endpoints of the interval. The smallest value obtained from these evaluations is the absolute minimum value of the function.
2. Graphical Method – this involves plotting the graph of the function and identifying the lowest point on the graph. The y-coordinate of that point is the absolute minimum value of the function.
It is important to note that the absolute minimum value may not always be unique, meaning that there could be multiple points in the domain of the function that produce the same minimum output value. In such cases, it is necessary to identify all of these points in order to fully describe the absolute minimum value.
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