Exercise 15

In this exercise we consider how to apply the Method of Lagrange Multipliers to optimize functions of three variable subject to two constraints. Suppose we want to optimize $f = f(x,y,z)$ subject to the constraints $g(x,y,z) = c$ and $h(x,y,z) = k$. Also suppose that the two level surfaces $g(x,y,z) = c$ and $h(x,y,z) = k$ intersect at a curve $C$. The optimum point $P = (x_0,y_0,z_0)$ will then lie on $C$.

1. Assume that $C$ can be represented parametrically by a vector-valued function $\vr = \vr(t)$. Let $\overrightarrow{OP} = \vr(t_0)$. Use the Chain Rule applied to $f(\vr(t))$, $g(\vr(t))$, and $h(\vr(t))$, to explain why $$\begin{aligned} \nabla f(x_0,y_0,z_0) \cdot \vr'(t_0) & = 0, \\ \nabla g(x_0,y_0,z_0) \cdot \vr'(t_0) & = 0, \text{ and } \\ \nabla h(x_0,y_0,z_0) \cdot \vr'(t_0) & = 0. \end{aligned}$$ Explain how this shows that $\nabla f(x_0,y_0,z_0)$, $\nabla g(x_0,y_0,z_0)$, and $\nabla h(x_0,y_0,z_0)$ are all orthogonal to $C$ at $P$. This shows that $\nabla f(x_0,y_0,z_0)$, $\nabla g(x_0,y_0,z_0)$, and $\nabla h(x_0,y_0,z_0)$ all lie in the same plane.

2. Assuming that $\nabla g(x_0,y_0,z_0)$ and $\nabla h(x_0,y_0,z_0)$ are nonzero and not parallel, explain why every point in the plane determined by $\nabla g(x_0,y_0,z_0)$ and $\nabla h(x_0,y_0,z_0)$ has the form $s\nabla g(x_0,y_0,z_0)+t\nabla h(x_0,y_0,z_0)$ for some scalars $s$ and $t$.

3. Parts (a.) and (b.) show that there must exist scalars $\lambda$ and $\mu$ such that $$\begin{gathered} \nabla f(x_0,y_0,z_0) = \lambda \nabla g(x_0,y_0,z_0)+ \mu \nabla h(x_0,y_0,z_0). \end{gathered}$$ So to optimize $f = f(x,y,z)$ subject to the constraints $g(x,y,z) = c$ and $h(x,y,z) = k$ we must solve the system of equations $$\begin{aligned} \nabla f(x,y,z) & = \lambda \nabla g(x,y,z)+ \mu \nabla h(x,y,z), \\ g(x,y,z) & = c, \text{ and } \\ h(x,y,z) & = k. \end{aligned}$$ for $x$, $y$, $z$, $\lambda$, and $\mu$.

   Use this idea to find the maximum and minimum values of $f(x,y,z) = x+2y$ subject to the constraints $y^2+z^2=8$ and $x+y+z = 10$.