Theoretical basis of mathematical analysis 22: Cauchy mean value theorem

Theorem: Set the function and satisfy:

1. continuous in all directions

2. It can be guided in all directions.

3. The sum is not zero at the same time

Then, make

Prove:

As an auxiliary function

Obviously meet the conditions of Rolle theorem.

Therefore, make

Will write a parametric equation with parameters.

Represents a curve on a plane.

Represents the slope of the chord connecting the two ends of the curve.

It represents the slope of the tangent at the corresponding point on the curve.

Indicates that the tangent and chord are parallel to each other.

1. Let the function be continuous and differentiable in the domain, then, let

Certificate:

set up

Obviously, the conditions of Cauchy mean value theorem are satisfied simultaneously in the field.

Therefore, make

Surface treatment can be carried out.

2. The set is differentiable on the interval, and it is proved that it is uniformly continuous on the interval.

Certificate:

set up

party history

rule

Through Cauchy mean value theorem

manufacture

Consistent and continuous in the world

And it is continuous in the world, so consistent and continuous.

Uniformly continuous in interval