Question

Please help me solve the following problem using Lagrange multipliers. Find a vector in 3-space whose length is 5 and whose components have the largest possible sum.

Answer 1

Let \(\mathbb{v}=\langle x,y,z \rangle\). We want to maximize \(f(x,y,z)=x+y+z\), given that \(\sqrt{x^2+y^2+z^2}=5\). To use Lagrange multipliers, first rewrite the constraint as \(g(x,y,z)=x^2+y^2+z^2-25\), then form and solve the system \(f_x=\lambda g_x\), \(f_y=\lambda g_y\),\(f_z=\lambda g_z\).