Need to find the minimum value of mathematical function represents practical interest in the solution of applied tasks, for example, in economy. Minimization of losses is of great importance for business activity.
Instruction
1. To find the minimum value of function, it is necessary to define at what value of an argument x0 inequality of y(x0) ≤ will be carried out by y (x) where x ≠ x0. As a rule, this problem is solved on a certain interval or in all area of values of function if that is not set. One of aspects of the decision is finding of stationary points.
2. A stationary point is called the value of an argument at which the derivative of function addresses in zero. According to Fermat's theorem if differentiable function accepts extreme value in some point (in this case – a local minimum), then this point is stationary.
3. Function often accepts the minimum value in this point, however it can be defined not always. Moreover, it is not always possible to tell with an accuracy what the minimum of function is equal to or it accepts infinitesimal value. Then, as a rule, find a limit to which she aspires at decrease.
4. To define the minimum value of function, it is necessary to execute the sequence of actions consisting of four stages: finding of a range of definition of function, receiving stationary points, the analysis of values of function in these points and on the ends of an interval, identification of a minimum.
5. So, let some y (x) function on an interval with borders in points And yes V. Naydite the field of its definition is set and find out whether the interval is its subset.
6. Calculate a function derivative. Equate the received expression to zero and find equation roots. Check whether these stationary points get to an interval. If is not present, then at the next stage they are not considered.
7. Consider an interval regarding type of borders: opened, closed, combined or infinite. Depends on it as you will look for the minimum value. For example, the piece [And, C] is the closed interval. Set up them in function and calculate values. Do the same with a stationary point. Choose the minimum result.
8. With open and infinite intervals the situation is slightly more difficult. Here it is necessary to look for unilateral limits which not always yield unambiguous result. For example, for an interval with one closed and one pricked out by border [And, C) it is necessary to find function at x = And yes a unilateral limit of lim y at x → V-0.