8—Multivariable Calculus 2298.12 Lagrange Multipliers
This is an incredibly clever method to handle problems of maxima and minima in several variables when there are
constraints.
An example: “What is the largest rectangle?” obviously has no solution, but “What is the largest rectangle
contained in an ellipse?” does.
Another: Particles are to be placed into states of specified energies. You know the total number of particles;
you know the total energy. All else being equal, what is the most probable distribution of the number of particles
in each state?
I’ll describe this procedure for two variables; it’s the same for more. The problem stated is that I want to
find the maximum (or minimum) of a functionf(x,y)given the fact that the coordinatesxandymust lie on
the curveφ(x,y) = 0. If you can solve theφequation foryin terms ofxexplicitly, then you can substitute it
intofand turn it into a problem in ordinary one variable calculus. What if you can’t?
Analyze this graphically. The equationφ(x,y) = 0represents one curve in the plane. The succession of
equationsf(x,y) =constant represent many curves in the plane, one for each constant. Think of equipotentials.
φ= 0f= 012 34f= 5φ= 0f= 012 34f= 5Look at the intersections of theφ-curve and thef-curves. Where they intersect, they will usually cross
each other. Ask if such a crossing could possibly be a point wheref is a maximum. Clearly the answer is no,
because as you move along theφ-curve you’re then moving from a point wherefhas one value to where it has
another.
The only way to havefbe a maximum at a point on theφ-curve is for them to touch and not cross. When
that happens the values offwill increase as you approach the point from one side and decrease on the other.
That makes it a maximum. In this sketch, the values offdecrease from 4 to 3 to 2 and then back to to 3, 4,
and 5. This point where the curvef= 2touches theφ= 0curve is then a minimum offalongφ= 0.