1. ## Lagrange Multipliers

The plane $\displaystyle x+y+z=4$ intersects the paraboloid $\displaystyle z=x^2+y^2$ in an ellipse. Find the points on the ellipse nearest to and farthest from the origin.

So is the goal here to just minimize and then maximize the distance formula $\displaystyle f(x,y,z)=\sqrt{x^2+y^2+z^2}$ with the constraints of $\displaystyle g_1:x+y+z=4$ and $\displaystyle g_2:z=x^2+y^2$?

2. Hello, davesface!

The plane $\displaystyle x+y+z\:=\:4$ intersects the paraboloid $\displaystyle z\:=\:x^2+y^2$ in an ellipse.
Find the points on the ellipse nearest to and farthest from the origin.

So is the goal to minimize and maximize the distance formula $\displaystyle f(x,y,z)\:=\:\sqrt{x^2+y^2+z^2}$
with the constraints of $\displaystyle g_1:x+y+z\:=\:4$ and $\displaystyle g_2:z\:=\:x^2+y^2$ ? . . . . Yes!

We have: .$\displaystyle F(x,y,z,\lambda,\mu) \;=\;\left(x^2+y^2+z^2\right)^{\frac{1}{2}} + \lambda(x+y+z-4) + \mu\left(z-x^2-y^2\right)$

Find the five partial derivatives, equate to zero, and solve the system.

. . $\displaystyle \begin{array}{ccccc}\dfrac{\partial F}{\partial x} &=& \dfrac{x}{\sqrt{x^2+y^2+z^2}} + \lambda - 2\mu x &=& 0 \\ \\ \dfrac{\partial F}{\partial y} &=& \dfrac{y}{\sqrt{x^2+y^2+z^2}} + \lambda - 2\mu y &=&0 \\ \\ \dfrac{\partial F}{\partial z} &=& \dfrac{z}{\sqrt{x^21+y^2+z^2}} + \lambda + \mu &=&0 \end{array}$

. . . . $\displaystyle \begin{array}{ccccc}\dfrac{\partial F}{\partial \lambda} &=& x + y + z - 4 &=& 0 \\ \\ \dfrac{\partial F}{\partial \mu} &=& z - x^2 - y^2 &=& 0 \end{array}$

I'll wait in the car . . .
.

3. I haven't ever seen someone take $\displaystyle \frac{\delta F}{d \lambda}$ or $\displaystyle \frac{\delta F}{d \mu}$ (and since those are constants, do those derivatives make sense?). According to my notes, it's set up as $\displaystyle \nabla f= \lambda \nabla g_1+ \mu \nabla g_2$, which I think for this problem would be:

$\displaystyle \left [ \begin{array}{cc} \frac{x}{\sqrt{x^2+y^2+z^2}} \\ \frac{y}{\sqrt{x^2+y^2+z^2}}\\ \frac{z}{\sqrt{x^2+y^2+z^2}} \end{array} \right ]= \lambda \left [ \begin{array}{cc} 1 \\ 1\\1 \end{array} \right ] + \mu \left [ \begin{array}{cc} 2x \\ 2y \\-1 \end{array} \right ]$