Section 14.8: Lagrange Multipliers

Method of Lagrange Multipliers To find the maximum and minimum values of \(f(x, y, z)\) subject to the constraint \(g(x, y, z) = k\) [assuming that these extreme values exist and \(\nabla g \neq \mathbf{0}\) on the surface \(g(x, y, z) = k\)]: (a) Find all values of \(x, y, z\), and \(\lambda\) such that \[ \nabla f(x, y, z) = \lambda \nabla g(x, y, z) \] and \[ g(x, y, z) = k \] (b) Evaluate \(f\) at all the points \((x, y, z)\) that result from step (a). The largest of these values is the maximum value of \(f\); the smallest is the minimum value of \(f\).

If we write the vector equation \(\nabla f = \lambda \nabla g\) in terms of components, then the equations in step (a) become \[ f_x = \lambda g_x \quad f_y = \lambda g_y \quad f_z = \lambda g_z \quad g(x, y, z) = k \] This is a system of four equations in the four unknowns \(x, y, z\), and \(\lambda\), but it is not necessary to find explicit values for \(\lambda\).

For functions of two variables the method of Lagrange multipliers is similar to the method just described. To find the extreme values of \(f(x, y)\) subject to the constraint \(g(x, y) = k\), we look for values of \(x, y\), and \(\lambda\) such that \[ \nabla f(x, y) = \lambda \nabla g(x, y) \quad \text{and} \quad g(x, y) = k \] This amounts to solving three equations in three unknowns: \[ f_x = \lambda g_x \quad f_y = \lambda g_y \quad g(x, y) = k \]