Mathematics for Economists

Calculus of variations

Example

Study the problem with kernelF



t,x,x



=

r

1 +



x

 2

.

This is a convex function as

d

du

p

1 +u^2 =

u

p

1 +u^2

d^2

du^2

p

1 +u^2 =

1

p

1 +u^2 up 1 u+u 2

1 +u^2 =

=

1 +pu^2 u^2

1 +u^2

1 +u^2 =

1

( 1 +u^2 )^3 /^2

> 0.