Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §13.7 Constraints and Lagrange Multipliers, #31 · Multivariable Calculus · Lagrange multipliers

Minimize … with … and … and

These problems allow inequality constraints, optional but good.

Minimize $f = 5 x + 6 y$ with $g = x + y = 4$ and $h = x \geq 0$ and $H = y \leq 0$ . Now $λ_{3} \leq 0$ and the sign change destroys Example 4. Show that equation (7) has no solution, and choose $x, y$ to make $5 x + 6 y < - 1000$ .

This exercise builds on another exercise in its source book, which isn't shown here.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §13.7 Constraints and Lagrange Multipliers, #31

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