Quiz 7, Complete Solution for Problem 4

The solution follows the steps in How to solve Max Min problems using Lagrange multipliers

I'll asssume thta you read the problem over; let's go to : Step 2:

We are trying to minimize total energy E, and the total energy depends on the currents x, y, and z. The only relation between the currents is that x+y+z=6.

Step 3
We know that the total energy is the sum of the energy in each branch, so

E=2x²+4y²+5z²

Step 4
We have one relation, x+y+z=6

This is already in the form of an equation for a level surface. If you compare this equation to g(x,y,z)=c, we see that x+y+z plays the part of g(x,y,z), while c=6.

For step 5 we try to solve the equations:

with f(x,y,z)=2x²+4y²+5z² and g(x,y,z)=x+y+z=6. When we plug in for f and g and write out the vector equation we get, we then want to solve:

so we get

Substituting for x , y and z into x+y+z=6

we get

so

This means that the total energy has a critical point when x=60/19, y=30/19 and z=24/19.

Step 6.

We know that we found the only critical point that the energy has in step 5. We just have to make sure the energy isn't smaller on the boundary. Think about the function f(x,y,z)=2x²+4y²+5z². The levels surfaces will be ellipsoids; as the value of f increases the ellipsoids get larger and larger. We get the following picture: when f=0, the level surface is just the origin, as the value of f increases the ellipsoid comes closer and closer to touching the plane wih equation x+y+z=6. Finally, the ellipsoid is tangent to the plane at x=60/19, y=30/19 and z=24/19.

As the value of f continues to increase, the ellipsoid intersects the plane in an elliptical curve. So f has its minimum value at the only point of tangency, and f increases as you move on the plane away from that point.

So the total energy is a minimum when x=60/19, y=30/19 and z=24/19.

Return to Quiz 7.