Extrema represent the maximum and minimum values of function and belong to its major characteristics. Extrema are in critical points of functions. And function in an extremum of a minimum and a maximum changes the direction according to the sign. According to definition, the first derivative from function is equal in a point of an extremum to zero or is absent. Thus, search of extrema of function consists of two tasks: findings of a derivative for the set function and definition of roots of its equation.
Instruction
1. Write down the set function f (x). Define its first derivative f’ (x). Equate the received expression of a derivative to zero.
2. Solve the received equation. Roots of the equation will be critical points of function.
3. Define what critical points - a minimum or a maximum - are the received roots. For this purpose find the second derivative f’’ (x) from initial function. Substitute in it in turn values of critical points and calculate expression. If the second derivative of function in a critical point is more than zero, then it will be a minimum point. Otherwise – a maximum point.
4. Count value of initial function in the received points of a minimum and a maximum. For this purpose substitute their values in expression of function and calculate. The received number will define a function extremum. And, if the critical point was a maximum, the function extremum will also be a maximum. Also in the minimum critical point the function will reach the minimum extremum.