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Advanced mathematics | proficient in mean value theorem problem solving routines summary

2022-06-25 02:37:00 【Xipi yo】

Problem solving routine summary

Routine one

notes ： In Cauchy mean value theorem g‘(x)≠0 Two denominators are limited at the same time You can use the counter evidence + Rolle

notes ： The monotonicity of a function cannot be judged by the positive or negative derivative at a point , But at least it can be judged that the function value of the de centered neighborhood of this point is the same as that of this point The size relation of the function value of the body . Specifically, it is the method of taking off the cap .

Use the method of taking off the cap to draw a conclusion , We can easily prove Fermat's theorem .

Class questions ： Please describe and prove Fermat's theorem 、 Derivative zero theorem 、 Derivative intermediate value theorem ( Also called Darboux theorem )

The root obtained from Rolle's theorem must be a single root .

Routine two Auxiliary polynomial

【 notes 】 structure ${\color{Red} F(x)=f(x)-p(x)}$ Purpose , It mainly refers to those in the question stem “ uneven ” The function value and derivative value of are all “ Zeroing ”, Thus making Be able to use Rolle's theorem constantly , Greatly simplifies the problem ！
【 notes 】 When setting a function , If zero is known , give an example , For example, quadratic function ： Set to ${\color{Red} A(x-a)(x-b)}$

【 notes 】 Reviewing the problem-solving process of auxiliary polynomials, we know , because yes n Sub polynomial , And we end up with auxiliary functions Continuously find n Secondary derivative , So we get $F^{(n)}(\xi )=0$ , So the real impact 、 To determine the final conclusion , Only The coefficient of the highest order term in ！ As for those lower order terms, they all disappear in the process of derivation .
（ Of course , This does not mean that the coefficients of the lower order terms can be written casually , Because only When each coefficient in is accurate , There are so many zeros and stagnation points , It is possible to satisfy the conditions for the use of Rolle's theorem , You can't just because the coefficients of those lower order terms don't affect $F^{(n)}(\xi )=0$ , You think they're useless , Tool man is also a man ~）

【 notes 】 Special note , In this question and Although there are the same minimum values , But the abscissa of their minimum values is not necessarily the same . So we need Classification of discussion . This idea can be applied to 2007 In the postgraduate entrance examination in ！

【 notes 】
As mentioned above , When using auxiliary polynomials to solve problems , If the conclusion is $f^{(n)}(\xi )=k$ , You need to construct a Polynomial fitting . because n Polynomial of degree has A coefficient of , So the question is in the stem Only When there are two independent conditions , Then all the coefficients in the auxiliary function cannot be solved smoothly .
To solve this problem , We need to give “ Impose constraints ”, Make the constructed
Can solve the problem smoothly .
As for what conditions need to be attached , You need to “ Specific analysis of specific problems ”, Can't generalize , But the core principle is “ What's missing is what's missing ”.

Supplementary examples ：

【 answer 】

It's easy to think of Rolle's theorem when we see that the title gives us three points

But it is found that these three points are not equal , Then we immediately think of Taylor's theorem

However, the intermediate value theorem of derivative cannot be directly used in the mathematics of postgraduate entrance examination ( Pay attention to the stem of this question [ No continuous ]), So what can we think of to avoid it ？

Construct auxiliary polynomials ：

【 The real question 】

Set three Formula method （ Integral factor method ）

【 Be careful 】
1、 Formula $\int g(x)dx$ No need to add any constant , Because we only need to find an auxiliary function .
2、 $F'(\xi )=0$ It is not necessarily the conclusion to be proved , Sometimes it takes a little more deformation , But be sure to pay attention to —— Yes $F'(\xi )=0$ When deforming , If both sides of the equation add something at the same time , Or divide something with it , It needs to be considered whether it is 0.

Need classified discussion :
①

If ξ≠0, It's in F'（ξ）＝0 Make an appointment on both sides ξ, Get the conclusion to be proved ;
②

If ξ=0, That means 0 In the interval （a,b） Inside , That is, within the definition domain .

Again because F（x）＝x²f（x）, therefore F（0）＝0, And because F（a）＝F（b）＝0, So to launch F There are three zeros , Namely a,0,b.

So we are 【a,0】 and 【0,b】 Use Rolle's theorem respectively , It can be proved that there are two non - 0 Of ξ1 and ξ2,

bring F'（ξ1）＝F'（ξ2）＝0, At this time due to ξ1 and ξ2 Not for 0, So you can be invited to .

【 notes 1】 The conclusion of this question is $f'''(\xi )=f(\xi )$ , The most intuitive feeling is “ First derivative Where is it ？”. therefore , This problem generally needs to be considered as the introduction of first-order derivatives , To reach a reduced order The effect of .
【 notes 2】 Subtract from both sides of the equation $f(\xi )$ , In fact, it can , It's essentially the same thing .
【 notes 3】 If the conclusion already contains , There is no need to introduce 了 , But sometimes we need to take the first derivative To distribute reasonably , Make them reach a kind of “ Balance ”.

To distribute reasonably , Reach a balance

【 notes 】 We will use this question as an example to show why in Mark's total Philosophy , Will emphasize “ Truth and error can be transformed into each other ”, It also hides the elements or sprouts of truth that will be revealed later .

In fact, the wrong solution is Lagrange from the inside out . The correct solution is from the outside to the inside Lagrange .

Set four Specific analysis of specific problems

Although we have already talked about the formula method , But formulas are not omnipotent , For this kind of problem, we need to observe the compact substructure , To analyze the result you want to prove is what the function looks like after deriving , Then try to restore it .

All in all , Restore the auxiliary function F(x) The process of entropy reduction is moving in the direction of entropy reduction , So it is normal to have difficulties ！

【 notes 】 This question should explain , Why doesn't the denominator be 0.

Pseudo double median problem ！

Routine five Separate constant and median , Direct pull or Cauchy

This topic is called the previous topic Example of routine 4 2

Routine six Double mean value theorem 【 False double median 】

【 notes 】 Take this topic as an example , Sum up a set of combination boxing , General killing ——
① Separate the complex median
② Restore it
③ An identical deformation or condition
④ Use the mean value theorem or Cauchy again
⑤ End of battle

【 Three median value example 】

Routine seven 【 The true double median theorem 】

【n Median problem 】

【 example 4： The three median values can also be taken from two intervals 】

【 Add 】 These different median values do not necessarily come from different intervals , It may also come from the interior of the open interval , The other comes from the end of the interval , This kind of topic is very interesting , For example, the following questions .

【 Before class questions Pave the way ~ Otherwise, you may not be able to find the subject conditions 】

Routine 8 Find the median θ The limits of

tricks Four parts ： Re expand , Compare the two forms 、 Establish the equation , Split median θ, Find the limit .

【 notes 】 Not clean , Two sides of an equation cannot be divided by the same .

【 example 4 The following question , Although it is also a calculation $\Theta$ The limits of , But because of f(x) The expression for is known , So we can easily solve the problem from the stem $\Theta$ The expression of , No difficulty 】

【 Wrong solution 】

【 The correct solution 】

Routine nine Taylor Mean Value Theorem

If you want to prove that the conclusion contains higher-order derivation ( Second derivative above )： In general, you can use “ Taylor expansion with Lagrangian remainder ” To prove . In the specific operation , The most important thing is to be appropriate Select the expansion point $x_{0}$ and Expanded point The location of .

generally speaking , Select the point with more derivative information as the expansion point , Select the point that only tells the function value as the expanded point .

tricks ：
- Finite interval ： Expand at the endpoint , Or expand at the midpoint
- Infinite interval ： Expand at any point

Be careful ：

【 notes 】
This problem needs to use Continuous function A corollary of the intermediate value theorem ：
$p,q>0\rightarrow p\cdot f(x_{1})+q\cdot f(x_{2})=(p+q)\cdot f(\xi )$
among $\xi\in [x_{1},x{_2}]$ , or $\xi\in [x_{2},x{_1}]$
because “ Derivative functions naturally satisfy the intermediate value theorem ”, So... In the condition “ The third derivative is continuous ” It can be changed to “ Third order differentiable ”！！

【 example 2 ： Extreme points contain information about derivatives , So the function is often expanded at the extreme point 】

notes ：

The next two questions , The stem condition does not tell the function value and derivative value at the experience point , Therefore, we can only guess how to select the expansion point and the expanded point according to the conclusion .

Experience ： stay “ In the middle ” an

example 4： On a straight road , A car starts and stops , Shared unit time and completed unit journey .. prove ： At least for a moment , The absolute value of its acceleration is not less than 4.

【 notes 】 For this kind of “ The function values of the two endpoints , Derivative values are known ” The topic , In order to make full use of the known conditions , Beginners may choose to expand each other at both ends , But the error will be very big （ Because Taylor expansion itself is an estimate , It requires that the distance between the expanded point and the expanded point should not be too large ）, So using The midpoint expands at the endpoint .

Take advantage of the above considerations , You can easily kill the following question .