absolute maximum
In mathematics, the concept of an absolute maximum refers to the largest value that a function can attain within its entire domain
In mathematics, the concept of an absolute maximum refers to the largest value that a function can attain within its entire domain. More formally, let’s consider a function f(x) defined on an interval [a, b]. The function has an absolute maximum at a point c, denoted as (c, f(c)), if and only if f(c) is the largest value that f(x) can take on within the interval [a, b].
To find the absolute maximum of a function, you generally follow these steps:
1. Identify the interval (domain) on which the function is defined. This is usually given in the problem or can be determined from the context.
2. Determine all critical points of the function within the interval. Critical points occur where the derivative of the function is either zero or undefined. To find critical points, set the derivative equal to zero and solve for x.
3. Evaluate the function at the critical points and at the endpoints of the interval. This includes finding the value of the function at the starting and ending points of the interval.
4. Compare the values obtained in step 3 and identify the largest value. This will be the absolute maximum of the function.
It is important to note that the existence of an absolute maximum is not guaranteed within every interval. A function may not have an absolute maximum if, for example, it is unbounded or oscillating within the domain.
Let’s demonstrate the process with an example:
Example:
Consider the function f(x) = x^2 – 4x + 5 defined on the interval [-1, 3].
1. The interval is given as [-1, 3].
2. To find critical points, we need to compute the derivative of f(x). The derivative of f(x) is f'(x) = 2x – 4. We set it equal to zero: 2x – 4 = 0. Solving for x, we find x = 2 as the critical point.
3. Evaluate the function at the critical point and the endpoints of the interval:
– f(-1) = (-1)^2 – 4(-1) + 5 = 1 + 4 + 5 = 10
– f(2) = (2)^2 – 4(2) + 5 = 4 – 8 + 5 = 1
– f(3) = (3)^2 – 4(3) + 5 = 9 – 12 + 5 = 2
4. Comparing the values obtained:
– The value at the critical point is f(2) = 1.
– The value at the starting point is f(-1) = 10.
– The value at the ending point is f(3) = 2.
We can see that the largest value obtained is 10, which occurs at x = -1. Therefore, the absolute maximum of the function f(x) on the interval [-1, 3] is ( -1, 10).
I hope this explanation helps you understand the concept of finding the absolute maximum of a function!
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