Absolute Min/Max
In mathematics, the terms “absolute minimum” and “absolute maximum” are used to describe the highest and lowest values, respectively, that a function can achieve on a particular interval or over its entire domain
In mathematics, the terms “absolute minimum” and “absolute maximum” are used to describe the highest and lowest values, respectively, that a function can achieve on a particular interval or over its entire domain.
Let’s consider a function f(x) defined on a closed interval [a, b].
1. Absolute Minimum: The absolute minimum of f(x) on the interval [a, b] is the lowest value that the function can attain on that interval. In other words, there is no other point on the interval where the function has a smaller value.
Mathematically, we say that c is an absolute minimum of f(x) if f(c) is the smallest value of f(x) on the interval [a, b]. It means that for any other point x on the interval, f(c) ≤ f(x).
2. Absolute Maximum: The absolute maximum of f(x) on the interval [a, b] is the highest value that the function can achieve on that interval. Similarly to the absolute minimum, there is no other point on the interval where the function has a larger value.
Mathematically, we say that d is an absolute maximum of f(x) if f(d) is the largest value of f(x) on the interval [a, b]. It means that for any other point x on the interval, f(d) ≥ f(x).
It is important to note that the absolute minimum and absolute maximum are not necessarily unique. A function can have multiple absolute minimum and maximum values on different intervals or even at the same point if the function is constant.
To find the absolute minimum and maximum of a function on a given interval, we need to take the following steps:
1. Determine all critical points of the function by finding where the derivative is equal to zero or undefined.
2. Evaluate the function at the critical points and at the endpoints of the interval.
3. Compare the values obtained to determine the absolute minimum and maximum.
It is also worth mentioning that the absolute minimum and maximum can sometimes be referred to as the global minimum and maximum, respectively, when considering the entire domain of the function.
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