Advanced Mathematics (Summary of several theorems of 1-derivative)

1) Fee Horse lemma:

The key to proof: if it is x0, then the left derivative at x0 equals the right derivative equal to the derivative.
Meaning: Obviously (x0,f (x0)) is an extreme point of f (x) in the neighborhood of X0, which promotes: the extreme point of f (x), which satisfies F ' (x) =0. The method of seeking the resident point is given.
2) Lowe's theorem

Proof point: The continuous interval exists maximum and minimum value, Fermat lemma;
Geometrical meaning: continuous interval [A, b], if the opening interval is conductive, and f (a) =f (b), there is at least one resident point within the interval.
3) LaGrand-Japanese value theorem:

Proof point: Construct auxiliary function g (x) =f (x)-F (a)-(X-A) (f (b)-F (a))/(B-a), Raul theorem.
Geometrical meaning: There must be a point on f (x), where the tangent of the point is parallel to AB.
Other meanings: The finite increment theorem can be introduced: F (x+δx)-f (x) =f ' (x+θδx) Δx ==δy=f ' (x+θδx) Δx.

4) Cauchy Middle value theorem

Proof points: Auxiliary functions and the Lowe theorem.

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Advanced Mathematics (Summary of several theorems of 1-derivative)