Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §13.7 Constraints and Lagrange Multipliers, #1 · Multivariable Calculus · Lagrange multipliers

Note: This direct approach _reduces_ to one unknown

Example 1 minimized $f = x^{2} + y^{2}$ subject to $2 x + y = k$ .
Solve the constraint equation for $y = k - 2 x$ , substitute into $f$ , and
minimize this function of $x$ . The minimum is at $(x, y) =$ ____,
where $f =$ ____.

Note: This direct approach _reduces_ to one unknown $x$ . Lagrange
_increases_ to $x, y, λ$ . But Lagrange is better when the first step of
solving for $y$ is difficult or impossible.

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, #1

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