Gradients & directional derivatives

What the problems look like

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

1. Standard

   Use Lagrange multipliers to find the maximum value of $f(x, y) = 1 xy$ subject to the constraint $x + y = 8$ with $x, y \ge 0$.

2. Standard

   Use Lagrange multipliers to find the maximum value of $f(x, y) = 1 xy$ subject to the constraint $x + y = 6$ with $x, y \ge 0$.

3. Standard

   Use Lagrange multipliers to find the maximum value of $f(x, y) = 2 xy$ subject to the constraint $x + y = 8$ with $x, y \ge 0$.

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