The equations (7.78) can be interpreted that when a member of the family of
curves f(x, y ) = c=constant is tangent to the constraint curve g(x, y ) = 0 , there results
the common values of
dy
dx
=
−∂f∂x
∂f
∂y
=
−∂x∂g
∂g
∂y
⇒
∂f
∂x
∂g
∂y
−
∂f
∂y
∂g
∂x
= 0.
One can give a physical picture of the problem. Think of the constraint condition
given by g=g(x, y ) = 0 as defining a curve in the x, y -plane and then consider the
family of level curves f =f(x, y ) = c, where c is some constant. A representative
sketch of the curve g(x, y ) = 0 ,together with several level curves from the family,
f=care illustrated in the figure 7-17. Among all the level curves that intersect the
constraint condition curve g(x, y ) = 0 select that curve for which chas the largest or
smallest value. Here it is assumed that the constraint curve g(x, y ) = 0 is a smooth
curve without singular points.
If (a, b) denotes a point of tangency between a curve of the family f =cand
the constraint curve g(x, y ) = 0, then at this point both curves will have gradient
vectors that are collinear and so one can write ∇f+λ∇g= 0 for some constant λ
called a Lagrange multiplier. This relationship together with the constraint equation
produces the three scalar equations
∂f
∂x
+λ∂g
∂x
=0
∂f
∂y
+λ∂g
∂y
=0
g(x, y ) =0.
⇒ ∂f
∂x
∂g
∂y
−∂f
∂y
∂g
∂x
= 0
(7 .79)
Lagrange viewed the above problem in the following way. Define the function
F(x, y, λ ) = f(x, y ) + λg (x, y ) (7 .80)
where f(x, y )is called an objective function and represents the function to be max-
imized or minimized. The parameter λ is called a Lagrange multiplier and the
function g(x, y )is obtained from the constraint condition. Lagrange observed that a
stationary value of the function F, without constraints, is equivalent to the problem
