Firstly, the discriminant of quadratic equation is used to find the extreme value.
When finding the extreme value of a certain kind of fractional function, if its numerator or denominator is a quadratic equation about X, it can be transformed into a univariate quadratic equation about X. According to X, there is a solution in the real number range, which can be found by discriminant.
Example 1. Find the function y= the extreme value of several methods to find the extreme value of the function.
Solution: transform the original function into a quadratic equation about x.
Several methods of finding the extreme value of function with (y- 1)x -2yx-3y=0.
∵x∈R and x≠3, x≠- 1,
∴: The above equation must have a solution in the real number range.
△= (-2y) Several methods for finding the extreme value of a function -4 (-3y)(y- 1)= 4y(4y-3)≥0
Several methods of finding the extreme value of function when y≤0 or y≥
Here, although there is no maximum (minimum) value of y, several methods for finding the extreme value of functions with y=0 and y= 0 correspond to x=0 and x=-3 respectively.
Therefore, when x=0, y has a maximum of 0, and when x=-3, y has a minimum. Several methods of finding extreme value of function.
Example 2: The range of several methods for finding the function y= finding the extreme value of the function.
Solution: transform the original function into: y+yx; Several methods for finding the extreme value of function =2x.
∵x∈R, ∴△= 4-4y Some methods for finding the extreme value of a function are ≥0, and the solutions are:-1≤y≤ 1.
∴ Function y= The range of several methods for finding the extreme value of a function is [- 1, 1].
From the above two examples, we can see that when using the discriminant of quadratic equation to find the extreme value of a function, Y is actually regarded as the coefficient of X, and the definition domain of the function is non-empty, that is, the equation has a solution, and the problem is transformed into solving a quadratic inequality. However, it should be noted that in the variational process, the value range of x can be expanded, but the extreme value of the function must be within the solution set of inequality. At this time, we should pay attention to the test, that is, take Y out of 2, whether X is meaningful, if not, we should abandon it and reconsider its extreme value.
Second, use the reciprocal relation to find the extreme value.
For some fractional functions, when the numerator contains no variables, the extreme value of the whole function can be obtained from the extreme value of the denominator.
Example 3, find the function y = 2-the minimum of several methods to find the extreme value of the function.
Solution: ∵x Some methods for finding the extreme value of a function -2x+6 = (x- 1) Some methods for finding the extreme value of a function +5 > 0.
∴ The definition fields of functions are all real numbers, and some methods of finding the extreme value of functions from X -2x+6=(x- 1) and some methods of finding the extreme value of functions +5 are known.
When x= 1, some methods of finding the extreme value of function, some methods of finding the extreme value of function by taking the minimum value,
∴ Several methods for finding the extreme value of a function: Some methods for finding the extreme value of a function take the maximum value,
At this time, y=2- some methods for finding the extreme value of the function take the minimum value 2- some methods for finding the extreme value of the function,
That is, when x= 1, the minimum value of y is 2-some methods to find the extreme value of function.
Third, using important inequalities to find the extreme value.
For a class of extreme values of functions with constant products and equal signs, we can consider using important inequalities to solve them.
Example 4: Some methods of finding the extreme value of function y=4x+.
Solution: Obviously, the domains of functions are all real numbers that are not equal to zero.
(1) When x > 0 and y = 4x+, several methods for finding the extreme value of a function ≥2 =2 = 12.
∴ Y has a minimum value of 12 when 4x = some methods for finding the extreme value of a function, that is, x = some methods for finding the extreme value of a function.
(2) when x < 0, let x = -t, then t > 0. Y = 4x+9/x =-(4t+ some methods for finding the extreme value of functions) ≤- 12.
∴ When x = some methods to find the extreme value of a function, y has a maximum value of-12.
When using important inequalities to solve problems, we must pay attention to the requirement that every term is positive. If all items are negative, we can extract a negative sign, so that each item in brackets is still positive. The above question is incomplete if only the first case is considered.
Example 5: Given L < 0 and M < 0, find the function y= the maximum of several methods for finding the extreme value of the function on (0, +∞).
Analysis: Although some methods for finding the extreme value of a function with x =8x = 2 are constants, some methods for finding the extreme value of a function with x = 8x = some methods for finding the extreme value of a function cannot solve the real number x, that is, there is no real number solution. Therefore, from y≥3 = 3.8 = 24, it is wrong to draw the conclusion that the minimum value of y is 24 by some methods of finding the extreme value of function. However, if we can divide some methods of finding the extreme value of the function from 8x and 64/x into equal m terms and n terms, and find out m, n and x by the design method, we can find out the minimum value of y. ..
Solution: Let y=x find some methods of function extremum+some methods of function extremum+…+some methods of function extremum+…+some methods of function extremum,
(Among them, some methods for finding the extreme value of a function have m terms, while others have n terms).
That is, when m= several methods for finding the extreme value of a function and n= several methods for finding the extreme value of a function (several methods for finding the extreme value of a function with x = several methods for finding the extreme value of a function with x = several methods for finding the extreme value of a function), y has a minimum value.
Several methods of finding the extreme value of function from 2+ = 3. Several methods of finding function extreme value (x methods of finding function extreme value x methods of finding function extreme value =x methods of finding function extreme value) get x methods of finding function extreme value +4x=96, and the only positive solution of this equation is x=2.
At this time, m = 4 and n = 2. At that time, the minimum value of y was 4+ 16+8=28 (substitution calculation).
Find the extreme value of function Y≥7. Some methods = 7. Several methods of finding the extreme value of function = 7.4 = 28.
Fourthly, method of substitution is used to find the extreme value.
Some irrational functions cannot be solved by the above methods. At this time, you can try to solve them by substitution.
Example 6. Several methods of finding the function y = finding the extreme value of the function-the maximum value of x in the interval [0, 1].
Solution: Let some solutions of function extreme value = t, then 0≤t≤ 1, x = t, some solutions of function extreme value.
∴ When t= several methods of finding the extreme value of function, namely, x= several methods of finding the extreme value of function, and several methods of finding the extreme value of function by taking the maximum value of y 。
In this paper, method of substitution is used to transform the form of irrational number into quadratic solution, which is a common method to find the extreme value of irrational number function. Especially for the function with the form of y=kx+, some methods to find the extreme value of t= function can be transformed into quadratic function, and then the extreme value can be found by matching method.
Example 7. Find the function y=x and some methods to find the extreme value of the function+1+2x (some methods to find the extreme value of the function 1-x) maximum and minimum values.
Solution: the domain of y is [- 1, 1], so x=cosθ(0≤θ≤π).
Several methods of finding the extreme value of function when y=
Some methods of finding the extreme value of function (some methods of finding the extreme value of function in y= are acute angles, and some methods of finding the extreme value of function)
∫- 1≤sin(2θ+α)≤ 1，
∴ Several methods of finding the extreme value of function ≤y≤ Several methods of finding the extreme value of function.
When sin (several methods for finding the extreme value of a function) =-1, several methods for finding the extreme value of a function.
Therefore, some methods to find the extreme value of the function of x =
Some methods of finding the extreme value of function by sin, and some methods of finding the extreme value of function by 2.
Therefore, some methods to find the extreme value of the function of x =
That is, when x =-, several methods to find the extreme value of function, and several methods to find the extreme value of function.
When x= some methods to find the extreme value of function, some methods to find the extreme value of function.
This question is based on the domain of the function [- 1, 1]. Thus, the irrational number function is transformed into a trigonometric function to solve the extreme value problem of the function.
Five, using analytical method to find the extreme value
Several methods to find the extreme value of y= formal function, in which (f(x) and g(x) are quadratic expressions about, and the coefficient of quadratic term is 1).
Extreme value, it is difficult to use pure algebra directly, because it has to be squared twice to remove the root sign. However, with the help of analytical methods, some methods for finding the extreme value of functions are regarded as the distance between two points in a plane rectangular coordinate system, which can be solved simply by using the properties of plane graphics.
Example 8. Find the function y= the minimum value of several methods to find the extreme value of the function, where a, b and c are all positive numbers.
Solution: Take points C (0, some methods for finding extreme value of function), D (c,-some methods for finding extreme value of function), M (x, 0) and B (c, 0) in rectangular coordinate system.
Then some methods of finding the extreme value of function =∣CM∣+∣MD∣.
That is, the sum of the distances from m to c and d.
According to the nature of plane figure, the sum of distances is the shortest if and only if the straight lines of C, M and D are three points * * *, and M is in the position of Mˊ.
∣ Om ∣: ∣ MB ∣ = ∣ OC ∣: ∣ BD ∣ from △ co m ∩△ DBM.
That is, some methods to find the extreme value of a function, and x= some methods to find the extreme value of a function.
At this time, some methods of finding the extreme value of function =∣CD∣= some methods of finding the extreme value of function.
Example 9. Find the function y= the range of several methods to find the extreme value of the function.
Analysis of several solutions of y= extreme value of function = several solutions of extreme value of function
Therefore, some methods of finding the extreme value of a function can be regarded as the difference between some methods of finding the extreme value of a function from point (x, 0) to point and some methods of finding the extreme value of a function from point in a plane rectangular coordinate system.
Solution: Take point A (-several methods of finding function extreme value, several methods of finding function extreme value), point B (several methods of finding function extreme value, several methods of finding function extreme value) and point M (x, 0) in rectangular coordinate system.
Then some methods of finding the extreme value of function =∣AM∣-∣BM∣.
Is the difference between the two sides of △ABM, which can be seen from the properties of plane graphics:
∣ AM ∣-∣ BM ∣ < ∣ AB ∣ = ∣ Several methods for finding the extreme value of a function ∣= 1
On the contrary, ∣ BM ∣-∣ AM ∣ < ∣ AB ∣ = 1.
∴∣y∣< 1
∴- 1< y < 1
This method is generally suitable for the sum-difference function of two quadratic roots, and the root number is quadratic function. At this time, it can be modified into the sum and difference of the distance between two points in the plane rectangular coordinate system by formula. This not only saves the trouble of square calculation, but also makes the formula have obvious geometric significance, which makes it easier to find out the solution and turn the difficult problem into a simpler problem. When solving the sum or difference of the distances from one point to the other two points on this axis, if the sum is an extreme value, then there is a minimum value when the connecting line of three points is * * *, that is, the distance between these two points. If it is a difference, there is no value, and the absolute value of the difference is less than the distance between these two points, the function value domain can be obtained.
Example 10. Find the function y= the range of several methods to find the extreme value of the function.
Analysis: This problem is not only a fractional function, but also a trigonometric function, which is often difficult to realize by pure algebra.
However, if we regard it as the slope of a straight line between a point (several methods to find the extreme value of a function) and a point (3,2), it is not difficult to solve it.
Solution: Let xˊ= several methods for finding the extreme value of function, and yˊ= several methods for finding the extreme value of function, then y= several methods for finding the extreme value of function.
That is, the slope of the straight line where the midpoint of the plane rectangular coordinate system (several methods to find the extreme value of the function) and (3, 2) are located,
(xˊ, yˊ)xˊ+yˊ= 1,
Therefore, it is only necessary to find the slope range of the connecting line between point (3, 2) and each point on the circle.
Let (3,2) and some methods of finding the extreme value of function with circle x+ and some methods of finding the extreme value of function with y = 1 The linear equation is
Yˊ-2=k (xˊ-3), that is, kxˊ-yˊ- 3k+2 = 0.
According to the distance formula from point to straight line, several methods to find the extreme value of function = 1,
That is, (-3k+2) methods for finding the extreme value of function = 1+k methods for finding the extreme value of function, and-12k+3 = 0 for finding the extreme value of function at 8k.
∴k= Several Methods of Finding Extremum of Function
∴ When some methods for finding the extreme value of a function ≤k≤ some methods for finding the extreme value of a function, a straight line intersects a circle.
That is, function y= the range of several methods for finding the extreme value of function is [several methods for finding the extreme value of function, several methods for finding the extreme value of function]
The form f(x) = the range of several methods for finding the extreme value of a function, which can be regarded as the slope of a point on the plane (several methods for finding the extreme value of a function) and (-b, -d), which must be on the conic = 1, and then the point (-b) can be used. From the above example, we can see that in
The relationship between surface functions can also be regarded as: finding the maximum and minimum values of ternary functions and multivariate functions.
We already know the steps to find the maximum and minimum of a univariate function, and the same steps can also be used to find the maximum and minimum of a multivariate function. Below we give the steps to solve the maximum and minimum values of multivariate functions in practical problems. As follows:
A) Establish a functional relationship according to practical problems and determine its definition domain;
B): find the stagnation point;
C): Determine the maximum and minimum values in combination with practical significance.
Example: Find a point on the plane 3x+4y-z=26 to make it the shortest from the origin of coordinates.
A): First, establish the functional relationship and determine the domain.
Solving a problem with the shortest distance from the origin is equivalent to solving a problem with the smallest square distance from the origin. But the point p lies on a given plane, so z=3x+4y-26. Substituting it into the above formula, we can get the functional relationship we need:
-∞