Lagrange Multipliers?

Minimizing or maximizing the distance is equivalent to minimizing or maximizing its square. Consider the Lagrangean

L(x, y, z, λ1, λ2) = x^2+ y^2 + z^2 - λ1(x^2 + 2y^2 + 1 - z) - λ2(x -y + 2z - 4)

Its partial derivatives L1, L2, L3, L4, L5 with respect to each of the 5 variables, including the multipliers, are

L1(x, y, z, λ1, λ2) = 2x - 2xλ1 - λ2

L2(x, y, z, λ1, λ2) = 2y - 4yλ1 + λ2

L3(x, y, z, λ1, λ2) = 2z + λ1 - 2λ2

The derivatives with respect to the multipliers are just the symmetric of the cosntraints.

Now, you have to set this 5 derivatives to 0 and solve this system of 5 equations and 5 unknowns. Ann then test if each solution is a minimum, maximum or saddle point. For this, you'll may consider the Hessian of the augmented matrx, that is, the Hessian including the multipliers. But here we can see the curve C is compact, so thre will be a globa minimum and a global maximum (continuous funnction on compact set).

You finish up from here, solving these equations is a considerable algebraic work.