Lagrange Multipliers

Example 1

We would like to find the extrema of the function f(x,y) with the constraint that g(x,y) = 0.

Here is the surface plot with the elliptic paraboloid drawn as a wire frame and the points corresponding to g(x,y) = 0 colored in red.

If we draw just the points on the surface where g(x,y) = 0, we can see that there are maxima and minima on this curve.

Using Lagrange's Theorem, the extrema must occur where the gradients of f and g are parallel. Thus we must solve the following system of equations.

Example 2

A rectangular box is to be constructed from 12 square meters of cardboard. Find the dimensions of the box with the largest volume that can be constructed.

The dimensions of the rectangular box will have to be chosen off of the following surface.

Using Lagrange's Theorem,the extrema must occur where the gradients of v and s are parallel.Thus we must solve the following system of equations.

The first solution reported contains negative values for the dimensions, so it must be discarded. Here is the maximum valome of the rectangular box that can be constructed.

The location of the optimal box dimensions on the constraint surface is shown below. The constraint surface is drawn as a wire frame for clarity.

Example 3

Find the points on the sphere of radius 2 centered at the origin which are closest and farthest from the point with coordinates (3,1,-1).

Compare the sphere and the point (3,1,-1).

Here we can see the location of the point (3,1,-1) relative to the sphere of radius 2 centered at the origin.

Using Lagrange's Theorem,the extrema must occur where the gradients of d and g are parallel.Thus we must solve the following system of equations.

We can now see the locations of the closest and farthest points. The points of interest are shown in red and the sphere is drawn as a wire frame for clarity.

The distances from the closest and farthest point found are shown below.

Example 4

We want to find the extrema of the function f on the intersection of g and h.

The intersection of the cylinder defined by h and the plane defined by g is an elliptical curve. Thus we want to find the extrema of the plane defined by f on this elliptical curve. The figure immediately below shows just the plane given by g and the cylinder given by h.

The next figure shows the plane and cylinder above rendered as wire frames. Their intersection is shown as a thick red curve. The graph of the plane defined by f is also drawn.

Lastly we see just the intersection curve.

Using Lagrange's Theorem for locating the extrema of functions of several variables, we must solve the following system of equations.

The points where the function f takes on its maximum and minimum values on the elliptical curve are superimposed in the curve as black points.

The level sets of function f are planes, so by plotting these planes we can get some picture of the extrema of f on the intersection of g and h.