Lagrange Multipliers: Examples

1. Find the minimum value of the following function, subject to the constraint shown afterwards:

Solution:

Let . Then use the formula shown below to set up the system of equations:

The system is then:

The solution of this system is and Thus the optimum value is:

2. Let represent the temperature at each point on the sphere . Find the extreme temperatures on the curve formed by the intersection of the plane and the sphere.

Solution:

 The constraints are:

The formula for two constraints is:

Thus we can set of the following system of equations:

Solving the system of equations (try this on your own) tells us that or . If , then we get the critical points:

and . If then we get the critical points and .Thus, putting the points in T

tells us the temperatures at the four critical points:

Which tells us that 25 is a minimum and 30.33 is a maximum.