Mathematics for Economists

Calculus of variations

Example

Find the extremal solutions of

Z 1

0

exp(x)+tx dt ,x( (^0) )= 0 ,x( (^1) )=α.
The EulerñLagrange equation is
exp(x)= d
dt
t= 1.
Ifα=0 thenx(t)=0 is a solution, otherwise there is no solution. As the
kernel is a convex function, if there is an extremal solution then it is a
minimum.