1. matching method: the function of shape, and the maximum value of the function is determined according to the value of the extreme point or boundary point of the quadratic function.

2. Discriminant method: the fractional function in the form of is transformed into a quadratic equation about X with coefficient y, because it is easy to generate roots when finding the maximum value of Y, so it is necessary to check whether the corresponding value of X has a solution when finding the maximum value.

3. Using the monotonicity of the function, first define the domain and monotonicity of the function, and then find the maximum value.

4. Using mean inequality, shape function and ≥≤, pay attention to the application conditions of positive and definite. , that is, a and b are both positive numbers, both are fixed values, and whether the equal sign of a=b is established.

5. Substitution method: a function in the form of, so that x can be reversed, and the function about t can be obtained by substituting into the above formula. Pay attention to the range of the domain of t, and then find the maximum value of the function about t? There is also the triangle method of substitution and the parameter method of substitution.

6. The combination of numbers and shapes is like taking the left side of the formula as a function and the right side as a function, putting their images in the same coordinate system, observing their positional relationship, and using analytic geometry knowledge to find the maximum value. Find the maximum value of the shape with the slope formula of the straight line.

7. Find the maximum value of the function with the derivative. 2. Ask the domain to be symmetrical about the origin before judging the relationship between f(x) and f(-x): if f(x)=f(-x), even function; If f(x)=-f(-x), odd function.

For example, the function f (x) = x 3, the domain is r, and it is symmetrical about the origin; And f (-x) = (-x) 3 =-x 3 =-f (x), so f (x) = x 3 is odd function. Another example is: the function f (x) = x 2, the domain is r, and it is symmetrical about the origin; And f (-x) = (-x) 2 = x 2 = f (x), so f (x) = x 3 is an even function.

Extended data:

Generally speaking, the maximum value of a function is divided into the minimum value and the maximum value. Simply put, the minimum value is the minimum value of the function in the definition domain, and the maximum value is the maximum value of the function in the definition domain.

The geometric meaning of the maximum (minimum) value of a function-the ordinate of the highest (low) point of the function image is the maximum (minimum) value of the function.

minimum value

Let the domain of the function y=f(x) be I, and if there is a real number m, it satisfies the following requirements: ① For any real number x∈I, there is f(x)≥M, and ② there is x0 ∈ i. Let f (x0)=M, then we call the real number m the minimum value of the function y=f(x).

highest

Let the domain of the function y=f(x) be I, and if there is a real number m, it satisfies the following requirements: ① For any real number x∈I, there is f(x)≤M, and ② there is x0 ∈ i. Let f (x0)=M, then we call the real number m the maximum value of the function y=f(x). ?

linear function

Linear functions, also known as linear functions, can be represented by straight lines on the x and y axes. When the value of one variable in a linear function is determined, the value of another variable can be determined by a linear equation of one variable.

Therefore, is it a proportional function, that is, y=ax(a≠0)? It is also an ordinary linear function, that is, y=kx+b (k is an arbitrary constant that is not 0, and b is an arbitrary real number), as long as x has a range, that is, z.

When a<0 o'clock

When a<0, then Y decreases with the increase of X, that is, Y is inversely proportional to X, then Y is the smallest when X is the largest and the largest when X is the smallest. Example:

2≤x≤3, y is the smallest when x=3 and the largest when x=2.

When a>0 o'clock

When a>0, then Y increases with the increase of X, that is, Y is proportional to X, then when the value of X is the largest, Y is the largest, and when X is the smallest, Y is the smallest. Example:

2≤x≤3, y is the largest when x=3, and y is the smallest when x=2? [3]?

quadratic function

Generally, we call a function in the form of y = ax 2+bx+c (where a, b and c are constants and a≠0) as a quadratic function, where a is called a quadratic coefficient, b is a linear coefficient and c is a constant term. X is the independent variable and y is the dependent variable. The maximum number of independent variables to the right of the equal sign is 2.

Note: "Variable" is different from "unknown", so it cannot be said that "quadratic function means that the polynomial function with the highest number of unknowns is quadratic".

"Unknown" is just a number (the specific value is unknown, but only one value is taken), and "variable" can take any value within a certain range. The concept of "unknown" is applied in the equation (both functional equation and differential equation are unknown functions, but the unknown and unknown functions generally represent a number or function-special cases will also be encountered).

But the letters in the function represent variables, and their meanings are different. From the definition of function, we can also see the difference between them, just as function is not equal to function relationship.

The maximum value of quadratic function, like linear function, is related to a.

When a<0, the image opens when y = 2x&; sup2y = & ampfrac 12； X & ampsup2 is the same, so at this time, Y has the maximum value, and Y has only the maximum value (the conclusion can be drawn by connecting the image with the quadratic function).

At this point, the y value is equal to the y value of the vertex coordinates.

When a>0, the image is opened at y =-2x&; sup2y =-& amp； frac 12； X & ampsup2 is the same, so at this time, Y has a minimum value, and Y has only a minimum value (you can draw a conclusion by connecting the image with the quadratic function).

At this point, the y value is equal to the y value of the vertex coordinates.

References:

Baidu encyclopedia-function maximization