For functions f(x, y, z) = xyz and $g(x, y, z)=x^{2}+2 y^{2}+3 z^{2}-1$, write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes f subject to the constraint g(x, y) = 0.

If we want to find a point $P(x,y,z)$ that maximizes or minimizes the function $f(x,y,z)$ subject to the constraint $g(x,y,z)=0$ we have to find $x,\ y,\ z$ and $\lambda$ that satisfy the following Lagrange multipliers conditions:

$$\begin{align*} \nabla f(x,y,z)=\lambda \nabla g(x,y,z)\quad\text{and}\quad g(x,y,z)=0. \end{align*}$$

So first we need to find the gradients of the functions $f(x,y,z)$ and $g(x,y,z).$

$$\begin{align*} f_x(x,y,z)&=\frac{\partial}{\partial x}(xyz)=yz;\\ f_y(x,y,z)&=\frac{\partial}{\partial y}(xyz)=xz;\\ f_z(x,y,z)&=\frac{\partial}{\partial z}(xyz)=xy;\\ g_x(x,y,z)&=\frac{\partial}{\partial x}(x^2+2y^2+3z^2-1)=2x;\\ g_y(x,y,z)&=\frac{\partial}{\partial y}(x^2+2y^2+3z^2-1)=4y;\\ g_z(x,y,z)&=\frac{\partial}{\partial z}(x^2+2y^2+3z^2-1)=6z;\\ \Longrightarrow\quad \nabla f(x,y,z)&=\left<f_x,f_y,f_z\right>=\left<yz,xz,xy\right>\\ \nabla g(x,y,z)&=\left<g_x,g_y,g_z\right>=\left<2x,4y,6z\right>. \end{align*}$$

The equation $\nabla f(x,y,z)=\lambda \nabla g(x,y,z)$ can be rewritten as:

$$\begin{align*} f_x(x,y,z)&=\lambda g_x(x,y,z)\\ f_y(x,y,z)&=\lambda g_y(x,y,z)\\ f_z(x,y,z)&=\lambda g_z(x,y,z) \end{align*}$$

So the required Lagrange multipliers conditions are

$$\begin{align*} yz=2\lambda x\\ xz=4\lambda y\\ xy=6\lambda z\\ x^2+2y^2+3z^2-1=0. \end{align*}$$