First of all, the discussion on the classification of derivatives is mainly divided into two types:
The first one: discuss quadratic function.
1. Binomial coefficient.
Example 1: Set a function in which
(1) Discuss the number of extreme points of the function and explain the reasons;
(2) If the constant holds, the range of values.
(1) Re-discussion without general score: the consequences are a bit. . . . . . . .
Discussion:
(1): When,
So we just need to find another point in the interval to make it hold, and we can prove that there is an extreme point.
Looking for a point: when, therefore, there is;
Orders, solutions.
Therefore.
From Zero Theorem: Therefore, there is only one sign-changing zero in the interval.
Therefore, when the function has a maximum point.
(2): When, the function has no point.
(3): If the domain has a solution. Make the solution a reality.
.
Only the advantages and disadvantages of are discussed below.
A: Whenever it is, there is always a time when the function has no point.
B: Timely means timely; .
; So the conclusion is within the definition domain.
Let's start looking for some operations:
Find the left end:
Conditions: timely; Find some intervals:.
Verification:
.
suppose
.
Verification:.
From the zero theorem:
There is a sign zero in the interval.
So the interval has a maximum point.
Find the right ending:
Conditions: timely; Find some intervals:.
From the zero theorem:
There is a sign zero in the interval.
Therefore, there is a minimum point in the interval.
To sum up: f'(x) is in x >; There are two sign zeros in the-1 interval. Therefore, the function f(x) has two extreme points.
To sum up:
① If the function has a maximum point.
② If, the function has zero.
③ If, the function has two extreme points.
Summary:
The process shown above is logical, and it is hard to imagine:
The difficulty lies in two aspects:
Let's discuss it with quadratic function:
,
manufacture
Discussion:
(1): When, the function has no point.
(2): If only one sign-changing zero function has a maximum point.
(3) When and when the constant holds, the infinity point of the function.
(4) When,,, so there are two sign-changing zeros, that is, only two sign-changing zero functions have two extreme points.
To sum up:
① If the function has a maximum point.
② If, the function has zero.
③ If, the function has two extreme points.
It is obviously much easier to discuss quadratic functions after general division.
The second problem is to eliminate the parameters by exploring the necessary conditions and replacing the principal components.
Then when must there be a solution?
When it is timely, when it is timely, it must be solved.
2 points will be deducted for limiting writing, so why not? When adopted, there is always a point in the domain, so the provable range can only be in the interval.
Operation:
Conditions:.
We know:
therefore
manufacture
Solution: Within the definition domain.
So when, domain: when; There is always a point to make it come true. Therefore, if you want to make it, you must have it.
To sum up, you should be talented.
The following discussion only needs to be discussed in.
Replace principal components with independent variables and parameters:
Discussion:
(1) when, monotonically decreasing.
.
(2) When, the function can take any value.
(3) When monotonicity increases.
.
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