If f(x0)≠0, then |fx| is differentiable at x0 "==" fx is differentiable at x0

If f(x0)=0, |fx| can be derived at x0 as "==" f'(x0)=0

Principle (generalized use), in this derivative definition of fx, according to the derivative definition, x tends to a from both sides, but there is an absolute value, making the final result different (one positive g(a) and one negative g(a)) , if you want to make two limits that are opposite numbers the same, take both of them to be 0. That is, let g(a)=0. Generalize it to the function f(x)=g(x)|p(x)| and you can get it. If the real root of the absolute value function p(x)=0 is reflected in g(x)=0, then This point is derivable, otherwise it is not. In the absolute value, the points with px=0 are not differentiable, but because they are multiplied by gx=0 at this time, that is, the left and right derivative definition limits are equal and become differentiable.

Commonly used k|x| is not differentiable at 0, x|x| is differentiable at 0

This method can be used to determine the non-differentiable point of this type of function

Finding the number of roots can also be used to prove the existence of

The parameters are separated when deriving the derivation, and they will be eliminated after derivation to facilitate discussion.

$If f^{(n)}≠0 on the interval I, then f(x)=0 has at most n real roots$

It is often used to assist in proofs. If there are at least n roots calculated by hand and at most n roots calculated by Rolle, then there are n roots.

Rohr can also be used to determine the stationary point.

1. Analysis and restoration

2. Differential equations

3. Commonly used basic formulas (essentially also analytical reduction method)

4 Construct the function based on the inspiration given in the question

If one point is a, another point should also be a,

Definite integrals can also be used to evaluate, especially when a linear term is multiplied by a definite integral constant, 0 can be obtained directly, and then according to the integral mean value theorem, it can be concluded that there are points in the interval that satisfy this condition.

The twice-mean value theorem (pull the middle)

Cauchy and Lazhong

Determine segmentation point

Mean Value Theorem

Analyze the conditions in the question and perform Taylor expansion for x values ​​with more known forms. If several x have the same number of known parameters, find the x whose derivative value is known. If it is known that f(1)=0 and f'(2)=1, then expand 2 first.

Generally, Rolle is given priority, followed by Fermat. When two points with the same derivative = 0 cannot be found, Fermat is considered. Fermat often requires that the maximum value in a continuous interval must be the extreme value. This needs to be based on the meaning of the question. To find the extreme point, the derivative of the extreme point is 0, and this point is the point you are looking for.

Pulling in is also about slope

In any question that requires derivatives and functional relationships, Lagrangian can be considered, especially if the values ​​of some points are given.