Function. The order of finding the largest and smallest value of a continuous function on an open or

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5. Function. Necessary criteria for extremum.

The sequence of calculations to determine the smallest and largest function values in the open or infinite interval consists of the following steps.

Set whether the interval X is a subset function definition areas .

Select the set of points where the first one does not exist. derivative and which are located on the interval X (traditionally these points are found in functions with the argument under the sign of the modulus and power functions with a fractional rational index). When these points are not, then proceed to the next stage.

Set the set of stationary points located in the interval X. For this purpose, the derivative of the function is equated to zero, we find the roots the resulting equation and take only suitable. When there are no stationary points or none of them is in the interval, then proceed to the next stage.

We perform calculations of the function values ​​at stationary points and points at which the first derivative of the function does not exist (if there are such points).

As you can see, the sequence of performing actions up to this point was no different from finding the largest and smallest value of the function on the segment . Further, the course of calculations is determined by the interval X.

When interval X is characterized as:

(a; b) , calculate one-sided limits ;

(a; b] , set the value of the function at x = b and one-sided limit ;

[a; b) , set the value of the function at x = a and one-sided limit ;

(- ∞; + ∞), we make calculations limits by + ∞ and -∞ ;

[ a ; + ∞) , perform calculations of the value of the function at the point x = a and the limit at + ∞ ;

( a ; + ∞) , we calculate the one-sided limit and limit by + ∞ ;

(-∞; b ] set the value of the function at x = b and the limit at -∞ ;

(-∞; b ) find the one-sided limit and the limit is -∞ ;

Having obtained the values ​​of the function and limits, we carry out a sequential analysis. Many answers may be received. So, when the one-sided limit equals minus infinity (plus infinity), then o maximum (minimum) value of the function nothing can be said for the selected interval.

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