# Maxima and Minima of Functions of Two Variables

Locate relative maxima, minima and saddle points of functions of two variables. Several examples with detailed solutions are presented. 3-Dimensional graphs of functions are shown to confirm the existence of these points. More on Optimization Problems with Functions of Two Variables in this web site.

__Theorem__

Let f be a function with two variables with continuous second order **partial derivatives** f_{xx}, f_{yy} and f_{xy} at a critical point (a,b). Let

_{xx}(a,b) f

_{yy}(a,b) - f

_{xy}

^{2}(a,b)

_{xx}(a,b) > 0, then f has a relative minimum at (a,b).

b) If D > 0 and f

_{xx}(a,b) < 0, then f has a relative maximum at (a,b).

c) If D < 0, then f has a saddle point at (a,b).

d) If D = 0, then no conclusion can be drawn.

We now present several examples with detailed solutions on how to locate relative minima, maxima and saddle points of functions of two variables. When too many critical points are found, the use of a table is very convenient.

^{2} + 2xy + 2y^{2} - 6x
Solution to Example 1:Find the first partial derivatives f _{x} and f_{y}.
f _{x}(x,y) = 4x + 2y - 6
f _{y}(x,y) = 2x + 4y
The critical points satisfy the equations f _{x}(x,y) = 0 and f_{y}(x,y) = 0 simultaneously. Hence.
4x + 2y - 6 = 0 2x + 4y = 0 The above system of equations has one solution at the point (2,-1). We now need to find the second order partial derivatives f _{xx}(x,y), f_{yy}(x,y) and f_{xy}(x,y).f _{xx}(x,y) = 4
f _{yy}(x,y) = 4
f _{xy}(x,y) = 2
We now need to find D defined above. D = f _{xx}(2,-1) f_{yy}(2,-1) - f_{xy}^{2}(2,-1) = ( 4 )( 4 ) - 2^{2} = 12
Since D is positive and f _{xx}(2,-1) is also positive, according to the above theorem function f has a local minimum at (2,-1).The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6).
^{2} - 4xy + y^{4} + 2
A 3-Dimensional graph of function f shows that f has two local minima at (-1,-1,1) and (1,1,1) and one saddle point at (0,0,2).
^{4} - y^{4} + 4xy
A 3-Dimensional graph of function f shows that f has two local maxima at (-1,-1,2) and (1,1,2) and a saddle point at (0,0,0). Exercise: Determine the critical points of the functions below and find out whether each point corresponds to a relative minimum, maximum, saddle point or no conclusion can be made.
1. f(x , y) = x ^{2} + 3 y^{2} - 2 xy - 8x
2. f(x , y) = x ^{3} - 12 x + y^{3} + 3 y^{2} - 9y
Answer to Above Exercise:1. relative maximum at (1,1) and (-1,-1) and a saddle point at (0,0) 2. relative maximum at (2,-3), relative minimum at (2,1), saddle points at (-2,-3) and (-2,1). More on partial derivatives and mutlivariable functions. Multivariable Functions Home Page |