I have the following problem:

You have max(x^{2}+y^{2}) which his restrained on the set x^{2}+y^{2}=1

1) Prove that this problem has solution.

2) Solve the Lagrange problem

3) Find the optimal value of the objective function

Here my answers:

1) It has a solution because the function is continous and it's restraint is a closed and bounded set. Therefore the extrem value theorem tells us that it attains a minimum and a maximum value at least once.

2) L_{(x,y)}=x^{2}+y^{2}*z(x^{2}+y^{2}-1)

Now I take the three derivatives, set them zero and solve the system of equations, so I get the points:

(-+1,0) and (0,+-1) and (+-2^{1/2}/2,+-2^{1/2}/2)

So the first two points are the maximum while the third point is the minimum value.

3) I don't really understand the question here. What could be meant by optimal value?

I can see that the value of the function is 1 at its maximum and 1/2 at its minimum. - Could that be the answer?

Thank you for your help!