Assignment:
Consider the following function:
f(x,y) = xy
on the set S = {x^2 +4y^2 ≤ 1}.
a) Explain by applying a relevant theorem why f(x,y) has a global maximum and a global minimum in the set S.
b) Find the critical of f in the interior of the set S.
c) Use the method of Lagrange multipliers to find the minima and maxima of f on the boundary of S given by x^2 + 4y^2 =1.
d) Find the global maximum and the global minimum of f.
Provide complete and step by step solution for the question and show calculations and use formulas.