Lagrange multipliers

Problem 10.195 · medium

1. Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
   Reviewed
2. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x + 3 y\right)\\\frac{\partial}{\partial y} \left(2 x + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}2\\3\end{matrix}\right] \]
   ∇f.✓ Proved
3. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 25\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 25\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
   ∇g.✓ Proved
4. So x = 2/(2λ), y = 3/(2λ); substituting into the constraint gives λ = ±√(13)/(2√25).
   Reviewed
5. \[ 5 \sqrt{13} \]
   The maximum; the minimum is its negative.✓ Proved