Absolute Maximum 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.
In mathematics, the absolute maximum value is the largest possible value a function can attain. This value is considered absolute because it is the maximum value in the entire domain of the function. To find the absolute maximum value of a function, you need to find the highest point on its graph.
To find the absolute maximum of a function, you can follow these steps:
1. Find the critical points of the function by taking the derivative of the function and solving for the points where the derivative is equal to zero or undefined.
2. Determine which of these critical points are maximums by evaluating the second derivative at these points. If the second derivative is negative, the critical point is a maximum.
3. Evaluate the function at these critical points and at the endpoints of the domain.
4. The largest of these values is the absolute maximum of the function.
For example, let’s find the absolute maximum of the function f(x) = x^2 – 2x + 3 in the domain [0,5].
1. Taking the derivative of the function and setting it equal to zero, we get:
f'(x) = 2x – 2 = 0
x = 1 is the critical point.
2. Evaluating the second derivative at x = 1, we get:
f”(1) = 2 > 0
Therefore, x = 1 is the minimum point.
3. Evaluate the function at x = 0, x = 1, and x = 5:
f(0) = 3
f(1) = 2
f(5) = 18
4. The largest value is f(5) = 18, so the absolute maximum value of the function is 18 in the given domain [0,5].
More Answers:
Mastering Arithmetic Mean: The Basics And Importance Of Calculating Averages In Statistical AnalysisExploring Calculus: Antiderivatives And Their Applications
Discovering The Absolute Minimum Value Of A Function: Analytic And Graphical Methods Explained