How to find a point of a maximum and a minimum

How to find a point of a maximum and a minimum

Points of a maximum and minimum are points of an extremum of function which are on a certain algorithm. It is an important indicator at a function research. The point of x0 is a minimum point if for all x inequality of f (x) ≥ by f(x0) is carried out from a certain vicinity of x0 (for a maximum point fairly return inequality of f (x) ≤ f(x0)).


1. Find a function derivative. The derivative characterizes change of function in a certain point and is defined as a limit of the relation of increment of function to increment of an argument which tends to zero. For its location use the table of derivatives. For example, function y derivative = to x3 will be equal to y’ = to x2.

2. Equate this derivative to zero (in this case x2=0).

3. Find value of a variable of this expression. It will be those values at which this derivative will be equal to 0. For this purpose substitute in expression any figures instead of x at which all expression will become zero. For example: 2-2x2 = 0 (1-x) (1+x) = 0x1 = 1, x2 =-1

4. Apply the received values on a coordinate line and calculate the sign of a derivative for each of the received intervals. On a coordinate line points which are accepted to a reference mark are noted. To calculate value on intervals substitute any values suitable by criteria. For example, for the previous function to an interval-1 it is possible to choose value-2. On an interval it is possible to choose from-1 to 1 0, and for values more than 1 choose 2. Substitute these figures in a derivative and find out the sign of a derivative. In this case the derivative with x =-2 will be equal-0.24, i.e. is negative and on this interval there will be a minus sign. If x=0, then value is equal 2, so on this interval the positive sign is put. If x=1, then a derivative also is equal-0.24 and therefore minus is put.

5. If when passing through a point on a coordinate line the derivative changes the sign from minus for plus, then it is a minimum point and if from plus on minus, then it is a maximum point.

Author: «MirrorInfo» Dream Team