absolute max
the largest y-value of the function, over the given interval (use x-values where first derivative = 0 & endpoints)
In mathematics, the term “absolute maximum” refers to the largest value that a function can attain within a particular domain. More formally, let’s consider a function f(x) defined on a closed interval [a, b]. If there exists a point c within this interval such that f(c) is greater than or equal to f(x) for all x in [a, b], then f(c) is called the absolute maximum of f(x) on the interval [a, b].
To find the absolute maximum of a function, you need to evaluate the function at all critical points and endpoints within the given interval. Critical points are the points where the derivative of the function is either zero or undefined.
Here’s a step-by-step process to find the absolute maximum of a function:
1. Identify the domain: Determine the interval on which the function is defined.
2. Find all the critical points: Solve the equation f'(x) = 0 to obtain the critical points. Also, include any points where the derivative is undefined.
3. Evaluate the function at the critical points and endpoints: Substitute the critical points and endpoints into the function to get the corresponding function values.
4. Compare the function values: Determine the largest value among all the function values obtained in step 3. This largest value is the absolute maximum.
It’s important to note that not all functions have an absolute maximum. For example, if a function is unbounded (i.e., it continues to increase or decrease without any limit), then it does not have an absolute maximum on the given interval.
Additionally, the absolute maximum is unique if it exists. In other words, there can only be one largest value within the specified interval.
By finding the absolute maximum, you can better understand the behavior of a function and identify its highest point within a given range.
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