Math 16B – F05 – Supplementary Notes 3
The Lagrange Multiplier Method
The method of Lagrange multipliers is often effective in finding solutions of constrained ex-
tremum problems. In the two-variable version of such a problem, one is given a function f(x, y),
and one wishes to maximize it or minimize it under the constraint that another function g(x, y)
vanishes (i.e., one wishes to find a maximum or minimum of fon the level curve g(x, y) = 0).
As explained in our textbook (where you will also find examples), Lagrange’s method proceeds as
follows. One introduces a third variable λ(traditionally called a Lagrange multiplier), and one
defines a function F(x, y, λ) of three variables by
F(x, y, λ) = f(x, y) + λg(x, y).
The basic theorem underlying the method states that if f(x, y) attains a maximum or a minimum
at the point (a, b) under the constraint g(x, y) = 0, then there is a value cof λsuch that (a, b, c) is
a critical point of F:
(1) ∂F
∂x (a, b, c) = 0,∂F
∂y (a, b, c) = 0,∂F
∂λ (a, b, c) = 0.
Thus, in principle, one can find the candidates for the desired constrained extremum of fby solving
the three simultaneous equations (1) for a, b, c. In the nicest situations there will be only one
solution, which gives immediately the sought-for extremum (a, b) of f.
The aim here is to explain the geometric underpinning of the method. So assume f(x, y) does
have a maximum or a minimum at (a, b) under the constraint g(x, y) = 0. We shall assume further
that (a, b) is a critical point of neither fnor g, the most common case. Note first that the partial
derivatives of Fare given by
∂F
∂x =∂f
∂x +λ∂y
∂x ,∂F
∂y =∂f
∂y +λ∂g
∂y ,∂F
∂λ =g.
The third equality in (1), therefore, just says that g(a, b) = 0, i.e., that (a, b) satisfies the constraint.
The other two equalities in (1) can be written as
(2) ∂f
∂x (a, b) = −c∂g
∂x (a, b),∂f
∂y (a, b) = −c∂g
∂y (a, b).
What do these mean?
To shorten the notation, let’s define
α=∂f
∂x (a, b), β =∂f
∂y (a, b),˜α=∂g
∂x (a, b),˜
β=∂g
∂y (a, b).
Rewritten in the new notation, (2) becomes
(3) α=−c˜α, β =−c˜
β.
Suppose for definiteness that (a, b) is a maximum of f(x, y) under the constraint g(x, y) = 0,
and let m=f(a, b). Consider the level curve f(x, y) = m(see Figure 3.1). It separates the
region where fis larger than mfrom the region where fis smaller than m. On the level curve
g(x, y) = 0 the function ftakes no value larger than m, so that curve, although it touches the level
