Begin2.DVI

condition for f=f(x, y )to have an extremum value at a point (a, b)requires that the

differential df = 0 or

df =∂f ∂x

dx +∂f ∂y

dy = 0. (7 .75)

Whenever the small changes dx and dy are independent, one obtains the necessary

conditions that

∂f

∂x = 0 and

∂f ∂y = 0

at a critical point. Whenever a constraint condition is required to be satisfied,

then the small changes dx and dy are no longer independent and one must find the

relationship between the small changes dx and dy as the point (x, y )moves along the

constraint curve. From the differential relation dg = 0 one finds that

dg =∂g ∂x

dx +∂g ∂y

dy = 0

must be satisfied. Assume that ∂g∂y = 0, then one can obtain

dy =

−∂x∂g ∂g ∂y

dx (7 .76)

as the dependent relationship between the small changes dx and dy.

Figure 7-17.

Maximum-minimum problem

with constraint.

Substitute the dy from equation (7.76) into the equa-

tion (7.75) to produce the result

df =

1 ∂g ∂y

(∂f

∂x

∂g ∂y

−

∂f ∂y

∂g ∂x

) dx = 0 (7 .77)

that must hold for an arbitrary change dx. This

gives the following necessary condition. The critical

points (x, y ) of the function f, subject to the con-

straint equation g(x, y ) = 0, must satisfy the equa-

tions

∂f ∂x