How to find the minimum value of the function

To find the minimum value of the function, you need to determine what value of the argument x0 to run the inequality y(x0) ≤ y(x), where x ≠ x0. Usually this problem is solved at a certain interval or in the whole region of values of the function, if one is not specified. One aspect of the solution is the location of the stationary points.

A stationary point is called a value argument, in which the derivative of the function vanishes. According to the Fermat's theorem if a differentiable function takes the extreme value at some point (in this case a local minimum), then this point is stationary.

The minimum value of a function often accepts it at this point, however, it is possible to determine not always. Moreover, it is not always possible to say with certainty what is the minimum of the function or it takes an infinitely small value. Then, as a rule, find the limit to which it tends when descending.

In order to determine the minimum value of the function, you need to perform a sequence of actions, consisting of four stages: finding domain of a function, obtaining stationary points, analysis values of the function at these points and at the ends of the interval, identifying a minimum.

So, imagine you are given some function y(x) on the interval with boundary points A and B. Find the area of its definition and find out whether the interval is its subset.

Calculate the derivative of the function. Paranaita the resulting expression to zero and find the roots of the equation. Check whether these stationary points in the interval. If not, then at the next stage, they are ignored.

Consider the interval subject to boundary types: open, closed, combined or infinite. It depends on how you look for the minimum value. For example, the interval [A, b] is a closed interval. Substitute them into the function and calculate values. Do the same with a stationary point. Select the minimum result.

With open and infinite intervals the situation is more complicated. There will have to find one-sided limits that do not always give an unambiguous result. For example, for the interval with one closed and one-punctured boundary of [A, b) should find the function at x = A and one-sided limit lim y as x → -0.