## Minimum of a Function

### The minimum of a function refers to the lowest value that the function can take on within a given interval or the entire domain

The minimum of a function refers to the lowest value that the function can take on within a given interval or the entire domain. In other words, it represents the point on the graph of the function where it reaches its lowest point.

To find the minimum of a function, we typically look for critical points, which are points where the derivative of the function is either zero or undefined. Critical points can occur at the endpoints of the interval or at the points where the derivative equals zero.

Once we have identified the critical points, we analyze their values and determine which one corresponds to the minimum. If the critical point is a local minimum, it will have a lower value than the surrounding points. If it is a global minimum, it will have the lowest value among all points in the entire domain.

In addition to critical points, we may need to consider endpoints, which are the values that define the bounds of the interval. The function may have its minimum at one of the endpoints, or the minimum could occur within the interior of the interval.

To determine the minimum of a function, we evaluate the function at all critical points, as well as the endpoints if applicable. The smallest value obtained will represent the minimum of the function.

It’s important to note that the minimum of a function may not always exist. For example, if the function is unbounded from below, it will not have a minimum. Additionally, in some cases, the function might have a minimum at a point that is not a critical point, such as in a piecewise-defined function. In such cases, we would need to consider all possible points where the minimum might occur.

Overall, finding the minimum of a function requires identifying critical points, evaluating the function at these points, and considering the endpoints if applicable. Through this analysis, we can determine the lowest value that the function can achieve within a given interval or domain.

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