## Absolute maximum

### The absolute maximum of a function occurs at a certain point within its domain where the function takes the largest value among all the points in its entire domain

The absolute maximum of a function occurs at a certain point within its domain where the function takes the largest value among all the points in its entire domain. In simpler terms, it is the highest point on the graph of the function.

To determine the absolute maximum of a function, we need to consider all the critical points and the endpoints of the domain. A critical point is a point where the derivative of the function is either equal to zero or does not exist. These points could potentially be local maximums or minimums. We evaluate the function at these critical points as well as at the endpoints of the domain and compare their values to find the absolute maximum.

For example, let’s consider the function f(x) = 2x^3 – 3x^2 – 12x + 5 on the interval [-2, 3]. To find the absolute maximum, we follow these steps:

1. Find the derivative of the function:

f'(x) = 6x^2 – 6x – 12.

2. Find the critical points:

To find critical points, set f'(x) = 0:

6x^2 – 6x – 12 = 0.

Factor the quadratic equation: (2x + 2)(3x – 6) = 0.

Solve for x: x = -1, x = 2.

3. Evaluate the function at the critical points and the endpoints of the interval:

f(-2) = 51, f(-1) = -16, f(2) = -11, f(3) = 38.

4. Compare the values and identify the absolute maximum:

From the results above, we can see that the largest value is f(-2) = 51. Therefore, the absolute maximum of the function on the interval [-2, 3] is 51.

It’s important to note that not all functions may have an absolute maximum. For instance, a function that goes to infinity without any bounds won’t have an absolute maximum.

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