Mathematics for Economists

Calculus of variations

φa(x(a))+φb(x(b))+

Zb

a

F



t,x,x



dt! maxmin

The argument is very similar. Now letybe an arbitrary (di§erentiable)

function and let

ψ(λ) $ φa(x(a)+λy(a))+φb(x(b)+λy(b))+

+

Zb

a

F



t,x(t)+λy(t),x(t)+λy(t)



dt.

If the functionxis optimal then by Fermatís principleψ^0 ( (^0) )= 0.