# Conjugate points of extermals of functional

In summary, conjugate points of extermals of functional are pairs of points on a curve or surface where the tangent lines are parallel to each other. They are closely related to extremal problems and play a significant role in mathematical optimization by aiding in the determination of optimal solutions and the behavior of a functional at critical points. These points are calculated using the second variation of a functional, and they can also exist in higher dimensional problems, where they refer to points where the tangent hyperplanes are parallel. This concept can be extended to higher dimensions, making it applicable to a wide range of optimization problems.
Show that the extermals of any functional of the form integ (a->b) F(x,y') dx have no conjugate points.

Not sure how to start this question, any help would be appreciated

anyone know how to do this?

One of the reasons we ask that people show what they have done is to understand exactly what their problem is. I understand from your title that you are working in the calculus of variations but even so there are many different definitions of "conjugate". Exactly what is your definition of "conjugate point"?

## What are conjugate points of extermals of functional?

Conjugate points of extermals of functional refer to pairs of points on a curve or surface where the tangent lines are parallel to each other. They play an important role in the theory of extremal problems.

## How are conjugate points of extermals of functional related to extremal problems?

Conjugate points of extermals of functional are closely related to extremal problems because they provide information about the behavior of a functional at critical points. They can help identify optimal solutions for these problems.

## What is the significance of conjugate points of extermals of functional in mathematical optimization?

Conjugate points of extermals of functional are significant in mathematical optimization because they allow for the determination of optimal solutions and the characterization of the behavior of a functional at these points. This can aid in the development of efficient algorithms for solving optimization problems.

## How are conjugate points of extermals of functional calculated?

Conjugate points of extermals of functional are calculated using the concept of the second variation of a functional. This involves taking the second derivative of the functional and setting it equal to zero, and then solving for the critical points.

## Can conjugate points of extermals of functional exist for higher dimensional problems?

Yes, conjugate points of extermals of functional can exist for higher dimensional problems. In this case, they refer to points where the tangent hyperplanes are parallel to each other. The concept of conjugate points can be extended to higher dimensions, making it applicable to a wide range of optimization problems.

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