## Endpoint Extrema

### In mathematics, an endpoint refers to the point at which a line segment, ray, or curve starts or ends

In mathematics, an endpoint refers to the point at which a line segment, ray, or curve starts or ends. It represents the extreme points of a mathematical object.

Endpoint extremum, also known as absolute extremum, is a concept used in calculus to find the highest or lowest value of a function within a given interval. More specifically, an endpoint extremum is the maximum or minimum value of a function that occurs at one of the endpoints of the interval, rather than at a point within the interval.

To find the endpoint extremum of a function, you need to evaluate the function at the endpoints of the interval and compare the values. The highest value will be the maximum endpoint extremum, while the lowest value will be the minimum endpoint extremum. It’s important to note that endpoint extremum only occurs if the function is defined and continuous at the endpoints of the interval.

For example, let’s say we have a function f(x) = 2x^2 – 3x + 1 defined on the interval [0, 5]. To find the endpoint extremum within this interval, we evaluate the function at the endpoints:

f(0) = 2(0)^2 – 3(0) + 1 = 1 (endpoint extremum)

f(5) = 2(5)^2 – 3(5) + 1 = 36 (endpoint extremum)

In this case, the absolute minimum endpoint extremum is 1, which occurs at x = 0, and the absolute maximum endpoint extremum is 36, which occurs at x = 5.

Endpoint extremum is useful for finding the optimal value of a function within a given interval, especially in applications of optimization where you want to maximize or minimize certain quantities.

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