6. Applications and Extensions

Advanced Optimization — Quiz

Test your understanding of advanced optimization with 5 practice questions.

Practice Questions

Question 1

Consider the function $f(x,y) = x^2 + y^2 - 4x - 6y$. You want to find the global minimum subject to the constraint $g(x,y) = x + 2y - 5 = 0$. What is the correct setup for the Lagrange multiplier equations?

Question 2

You are optimizing the function $f(x,y) = x^2 + 3y^2$ subject to the constraints $x^2 + y^2 = 1$ and $x + y = 0$. How many Lagrange multipliers are needed to solve this problem?

Question 3

Find the stationary points of $f(x,y) = x^3 + y^3$ subject to the constraint $x + y = 1$. Which of the following points is a candidate for a constrained extremum?

Question 4

For the function $f(x,y) = x^2 + y^2$ subject to the constraints $x^2 - y = 0$ and $x + y - 4 = 0$, what is the correct system of equations to solve for the Lagrange multipliers?

Question 5

You are optimizing $f(x,y,z) = x^2 + y^2 + z^2$ subject to the constraints $x + y + z = 1$ and $x^2 + y^2 = z^2$. What is the correct expression for the gradient of the constraints in the Lagrange multiplier setup?