Let

Defined by functions  with . Let  with  with norm  a) Prove that  it is frechet differentiable b) Solve the equation for all , and see that it is the only critical point ( where D is the frechet derivative ) c) Prove that  it is a local minimum

J(y) = S) (1 + (y'(x))2)-1dx = C10,1 y(0) = 0, y(1) = 1 , = (0) = y() = 0 - 2 u 0,11 n Co[0,1]] I(u) = J(x + (x = (c)n DI(u)h = 0 0n J(y) = S) (1 + (y'(x))2)-1dx = C10,1 y(0) = 0, y(1) = 1 , = (0) = y() = 0 - 2 u 0,11 n Co[0,1]] I(u) = J(x + (x = (c)n DI(u)h = 0 0n