Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

Critical Points and Relative Extrema

A point(ab, )in the domain of f(xy, )for which bothfabx( ,)=0 and

fay(),0b= is said to be a critical point of f.

If the first-order partial derivatives of f are defined at all points in some

region in the xy-plane, then the relative extrema of f in the region can occur

only at critical points.

However, not every critical point is a relative maximum or a relative minimum. A
critical point that is neither a relative maximum nor a relative minimum is called a
saddle point. The below procedure involving second-order partial derivatives is used
to decide whether a given critical point is a relative maximum, a relative minimum, or
a saddle point.

The second Partials Test

Suppose thatf(ab, )is a critical point of the functionf(xy, ). Let

()() ()

2

D=−fabfab fabxx ,,yy ⎡⎣ xy ,⎤⎦

IfD< 0 , then f has a saddle point at(ab, )

IfD> 0 , andfabxx(),< 0 , then f has a relative maximum at(ab, ).

IfD> 0 , andfabxx(),> 0 , then f has a relative minimum at(ab, ).

IfD= 0 , then test is inconclusive and f may have either a relative extremum

or a saddle point at(ab, ).

Example 1

Classify the critical point of the functionf(xy x y, )=^22 +.

(Answer: The critical point (0, 0)is the relative minimum.)

Example 2

Classify the critical point of the functionf(xy, )=y x^22 −.

(Answer: The saddle point(0, 0))

Example 3
Find all the local minima, local maxima, and saddle points for the function
f()xy x xy y,9=^22 −+−+x 5

Solution
Find the first partial derivatives fx and fy.

fxy()xy,2 9and ,=−−x y f xy( )=−+x 2 y

Now, to find any critical point, solve the following system.
290
6, 3
20

xy

xy

xy

⎧ −−=

⎨ ⇒= =

⎩−+ =

The only critical point is()6, 3.

The second partials are

fxyxx(),2, , 1and==−fxyxy( ) fxyyy( ,2)=