Topic 14. Optimization

Setting up one-dimensional optimization tasks - find the extremum (least or most) of the target function that is given on the set.

The Weierstrass Theorem. Any function that is continuous on a segment takes on its least and most important in this segment , and therefore there are points such that for any inequality £ £ for any one.

Points in which the derivative of a function equals zero are called critical or stationary points of a function). At the critical point, the "speed of the function" equals zero. The function can have the least (highest) value in one of the two boundary points of the segment, or in any of its internal points. In this case, the point must necessarily be critical - the necessary condition for the extremum.

In order to determine the smallest and largest value of the function f (x) on the segment [a, b], it is necessary to find all its critical points on the given segment, connect them to the boundary points of aib for all these points to compare the value of the function. The least and most of them give a solution to the optimization problem.

Example 1. Find the extreme values ​​of the function f (x) = 3x ⁴ -4x ³ -12x ² +2 on the segment [-2,3].

We find the derivative of the function f ¢ (x) = 12x ³ -12x ² -24x. To determine the critical points equate to zero the derivative function and find all the roots of the equation: 12x ³ -12x ² -24x = 0: x ₁ = -1, x ₂ = 0, x ₃ = 0. We compute the value of the function at these points, and also at the boundary points: f (-2) = 34, f (-1) = - 3, f (0) = 2, f (2) = - 30, f (3) = 29 A comparison of these numbers allows us to determine that the largest and smallest values ​​of the function at points respectively: fmin = (- 2) = - 30 i fmax (-2) = 34.

The method of uniformly dividing the points by segment

We take a certain number, and calculate the step and determine the value of the function in points. Among these numbers is the least. The number  can be taken at the smallest value of the function on the segment []. One of the problems is the definition of the number in order not to miss the extremum of the function.