🔬 Tutorial problems iota \(\iota\)#
Note
This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.
\(\iota\).1#
Solve the following constrained maximization problem using the Karush-Kuhn-Tucker method. Verify that the found stationary/critical points satisfy the second order conditions.
\[\begin{split}
\begin{array}{c}
f(x,y) = x^3 - y^3 \to \max_{x,y}\\
\text { subject to } \\
x^2 + y^2 \le 1,\\
x,y \in \mathbb{R}
\end{array}
\end{split}\]
\(\iota\).2#
Solve the following constrained maximization problem using the Karush-Kuhn-Tucker method. Verify that the found stationary/critical points satisfy the second order conditions.
\[\begin{split}
\begin{array}{c}
f(x,y) = -x^2 \to \max_{x,y} \\
\text { subject to } \\
x^2-y^2-2xy \ge 2,\\
x,y \in \mathbb{R}
\end{array}
\end{split}\]