table of Contents
Second, the application of the adjustment method
1. Multivariate functions and inequalities
Three, adjustment method and greed
One, adjustment method
In middle school mathematics, the adjustment method is widely used and can be used in all fields.
Its mathematical essence can be described as finding the maximum value of a multivariate function, and finding the maximum value point by analyzing the conditions satisfied by any two free variables at the maximum value point.
There are two adjustment directions in the adjustment method. One is to adjust the values of the two free variables to the smallest difference, and the other is to adjust the free variables to the largest difference.
Some people call it the local adjustment method and the polishing method.
Second, the application of the adjustment method
1. Multivariate functions and inequalities
For example, a+b+c=1, prove that f(a,b,c)=a^2+b^2+c^2 takes the minimum value when a=b=c
prove:
Assuming that a and b are not equal, adjust the values of a and b to (1-c)/2,
According to the basic inequality, the new function value is smaller.
So, when a=b=c, the function has a minimum
Such as:
prove:
The left side of the equation is abbreviated. For the right side of the equation, if z>=1/2 is also proved, the
If z<1/2, then yz+zx+xy-2xyz = z(x+y)+(1-2z)xy, the value of z is fixed, x+y=1-z is fixed, when x=y, get Minimum,
So, when x=y=z, there is a minimum value of 7/27
2. Geometry
For example, prove that the area of the equilateral triangle is the largest among the triangles with a certain perimeter
For example, in a figure enclosed by a closed curve with a constant perimeter, if there is a maximum area, it must be a circle.
3. Trigonometric function
For example, prove that in the acute triangle ABC, sin A + sin B + sin C> 2
prove:
The title is equivalent to proving that A, B, and C are all in the range of [0, PI/2], and the sum is PI, proving that sin A + sin B + sin C >=2, only when one of the angles is 0 To the minimum.
Since the sin function is convex on [0, PI/2], the greater the difference between A and B, the smaller sin A + sin B
Therefore, if sin A + sin B + sin C wants to get the minimum value, at most only one angle is not at the end of the interval.
Because the sum is PI, the minimum value is only taken when the three angles are PI/2, PI/2, and 0 respectively
Three, adjustment method and greed
Greedy is an algorithm that obtains the global optimal solution by solving the local optimal solution.
In terms of mathematics, the adjustment method and greed are essentially the same.
In terms of expression and calculation process, there are some differences between the adjustment method and greed.
Just like mathematical induction and dynamic programming, mathematical induction is generally to get a concise conclusion, dynamic programming is generally a cumulative calculation, and finally a calculation result is obtained.
The adjustment method generally also obtains a concise conclusion, and greed is generally a cumulative calculation, and finally a huge result is obtained, not just a numerical value.