7.3 Optimizing Functions of Two Variables

Visual – Page 557 - Transparency

Relative Extrema

The function f(x,y) has a relative maximum at the point P(a,b) in the domain of f if for all points (x,y) in a circular disk centered at P.

The function f(x,y) has a relative minimum at the point P(a,b) in the domain of f if for all points (x,y) in a circular disk centered at P.

Critical Points

If (a,b) in the domain of f and if both and then (a,b) is a critical point.

If a point is a relative extreme, it must be a critical point, but not all critical points are relative extrema.

Saddle Point

A saddle point is a relative maximum in one direction and a relative minimum in another direction.

The Second Partials Test

If f(a,b) is a critical point, then let

If D<0, then there is a saddle point at (a,b)

If D>0, and <0, then there is a relative maximum at (a,b)

If D>0, and >0, then there is a relative minimum at (a,b)

If D=0, the test is inconclusive.

Find the critical point and classify each as a relative maximum, a relative minimum, or a saddle point.

#2

#8

#6

#16

#18

Applications

Pg. 562 You can assume that a relative extremum you find as the solution to any practical optimization problem is actually the absolute extremum.

A company produces x units of commodity A and y units of commodity B. All the units can be sold for

p = 20 - 5x dollars per unit of A and q = 4 – 2y dollars per unit of B. The cost (in dollars) of producing these units is given by the joint-cost function . What should x and y be to maximize profit?