(1) Special position and limit position method: first consider the special position or limit position, determine the specific data of the maximum value, and then carry out reasoning proof under general circumstances. (2) Geometric theorem (axiom) method: apply the inequality properties and theorems in geometry. Common geometric properties are: the shortest line segment between two points; The vertical line from point to line is the shortest; The sum of two sides of a triangle is greater than the third side; Number-shape combination method with hypotenuse greater than right angle (3): analyze the algebraic relationship of problem variable elements, construct quadratic function, etc.
Algebraic maximum problems generally appear in the form of application problems, and common problems seek a scheme with the lowest cost, the least consumption, the highest output value and the greatest profit. As one of the compulsory examination questions for senior high school entrance examinations in various places, the difficulty is mainly intermediate, which is a compulsory examination question for all students. The key to solve this kind of problem is to establish a function model reasonably, set an unknown number reasonably on the basis of understanding the meaning of the problem, analyze the equivalence relationship in the problem, list the distinguishing function or equation, solve and discuss the meaning of the result, and end with "A: ……". It is important to note that if the listed equation is a fractional equation, it is necessary to test the increase of roots!
Specific example questions are as follows: