Surface area is:

Therefore, we want ........[1]

subject to the constraint

.....[2]

Solve [1] for z and sub into [2]:

....[3]

Differentiate [3] with respect to x and y:

...[4]

...[5]

Solve [4] for y and sub into [5]:

Solving this for x, we find

Then

and

Check to see if it is, indeed a minimum. Use the Second Partials Test.

Also,

Therefore, we have a relative minimum.

Now, LaGrange multipliers. See if we get the same thing.

Let

...[1]

...[2]

...[3]

From [1] and [2], we getx=y

From [2] and [3], we get

Sub these into the constraint, xyz, and we get:

Solving for y, we get

and

Same as before. Good. It checks.