Finding the Absolute Maximum of a Function | Step-by-Step Guide and Example

Absolute (or global) maximum

The absolute maximum is a concept in calculus that refers to the highest point on the graph of a function over a given interval or over its entire domain

The absolute maximum is a concept in calculus that refers to the highest point on the graph of a function over a given interval or over its entire domain. More formally, a function f(x) has an absolute maximum value at the point c if f(c) is the largest value of f(x) for all x in the domain.

To find the absolute maximum of a function, you need to evaluate the function at critical points and endpoints over the given interval or domain, and compare the values. The critical points are the points where the derivative of the function is zero or undefined.

Here’s a step-by-step process to find the absolute maximum of a function over a closed interval:

1. Find the critical points of the function by taking the derivative, setting it equal to zero, and solving for x. You should also check for any points where the derivative is undefined.

2. Evaluate the function at each critical point and calculate the corresponding y-values.

3. Evaluate the function at the endpoints of the interval and calculate the corresponding y-values.

4. Compare all the y-values obtained in steps 2 and 3. The largest y-value corresponds to the absolute maximum.

It is important to note that the absolute maximum may not exist if the function is unbounded or does not have a defined domain.

For example, consider the function f(x) = x^2 – 4x + 3 over the interval [0, 3].

1. Taking the derivative, we have f'(x) = 2x – 4. Setting it equal to zero, we get 2x – 4 = 0. Solving for x, we find x = 2, which is a critical point.

2. Evaluating the function at x = 2, we have f(2) = 2^2 – 4(2) + 3 = 3.

3. Evaluating the function at the endpoints, we have f(0) = 0^2 – 4(0) + 3 = 3 and f(3) = 3^2 – 4(3) + 3 = 0.

4. Comparing the y-values, we see that the absolute maximum is 3, which occurs at x = 0 and x = 2.

In summary, the absolute maximum of the function f(x) = x^2 – 4x + 3 over the interval [0, 3] is 3, attained at x = 0 and x = 2.

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