Lagrange multipliers

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Maximize $f(x, y) = xy$ subject to the constraint $g(x, y) = x + y + 3 = 0$. Using Lagrange multipliers ($\nabla f = \lambda \nabla g$), find the value of $\lambda$ at the optimal point.

2. Warm-up

   Maximize $f(x, y) = xy$ subject to the constraint $g(x, y) = x + y - 8 = 0$. Using Lagrange multipliers ($\nabla f = \lambda \nabla g$), find the value of $\lambda$ at the optimal point.

3. Warm-up

   Maximize $f(x, y) = xy$ subject to the constraint $g(x, y) = x + y - 7 = 0$. Using Lagrange multipliers ($\nabla f = \lambda \nabla g$), find the value of $\lambda$ at the optimal point.

Problems shown under their original terms: Generated (template, solver-computed).